Why do scientists think that all the laws of physics that apply in our galaxy apply in other galaxies? I like watching different videos about space. I keep seeing all these videos saying scientists found so and so at 200 billion light years away or this happened 13 billion years ago. 
My question is why do scientists think that all the physics that apply in our galaxy apply in a galaxy say 200 billion light years away?
What if, say at 135 billion light years away, all of a sudden the time space relationship changes drastically and instead of linear time space relationships the difference becomes based on a "sliding scale" (to revert back to high school). What if a light they first see and estimate to be 200 billion light years away has actually been traveling for another 300 billion light years before we could detect it? Lets be serious, we can't predict the weather farther out than 10 days accurately, and usually not that long....
 A: Because what we can observe doesn't differ according to distance from us.
The most important reason why we assume physical laws are not different in distant galaxies is that we can observe things in those galaxies that would not be as we observe them if the laws varied.
For example, we can see light from distant parts of the known universe. The spectral lines in that light tell us about the elements there. Those spectral lines tell us which elements are present, but we don't observe any new elements. And the elements we do observe seem to behave the same way they would in the sun or other nearby stars. Even small differences in the rules of physics would alter the emission or absorption lines of elements in ways we do not observe.
And we can do more than that. We can, to some extent, learn about the motion of distant objects from observing Doppler shifts in the spectral lines and these observations don't show us that the laws of gravity differ in distant places either.
I could go on, but the basic principle is that of the parts of the universe we can see the rules seem to be the same.
None of this says there cannot be unobservable things where the rules are different (but that is as much a philosophical problem as a scientific one). Some theories have suggested that that fundamental parts of physics do differ over time or in different parts of the universe, but they have all failed the key tests of science by not matching what we can observe.
A: 
What if, say at 135 billion light years away, all of a sudden the time space relationship changes drastically.

Well, it could.
Science is built on reasonable, well-founded assumptions, and a good scientist is open to the possibility of those assumptions being broken at some point in the future when more data becomes available. That is why no scientific statement is ever a 100% certainty, but a "theory". We must always be open to being unexpectedly proved wrong.
In this case, we have so far not encountered any evidence to suggest that the laws of physics are not universally applied, so for now we proceed on the assumption that they are. Because, otherwise, short of avoiding doing any science regarding far-away galaxies, what else could we do?
(It's a bit like how we only search for "life as we know it", not because we have ruled out the possibility of other, exotic kinds of life, but because what else would we do? How would we look for it? How would we recognise it? Any such thing would have to be discovered by accident.)
Again it is worth noting that all our observations so far support this assumption.
A: It might be considered a circular answer, but the ideas that we consider to be "laws" are precisely those ideas that are believed to be true everywhere. So if we discover that some phenomenon we've observed in our galaxy is not the same in Andromeda, the description of that phenomenon would not be a law.
What scientists do when they discover discrepancies like this is look for some more fundamental explanation that can be used to describe all the variations that have been observed. This might be a new equation with parameters that are specific to different locations (e.g. theories about galaxies might depend on the number of stars they contain and/or the age of the galaxy).
This has happened in the past, and not just in far off locations, and resulted in new laws being discovered. Mercury's orbit is not consistent with Newton's laws of gravitation. One of the results of Einstein's general theory of relativity was that it correctly explained this.
Scientists continue to consider modifications to their theories to explain new discoveries. For instance, some scientists have proposed that the speed of light has varied over the history of the universe, and this is an alternative explanation for the observations that led to cosmic inflation theory.
Put more generally, the history of cosmology has been a process of discovering how to describe the universe in more and more general ways, as our ability to observe it on larger scales and in smaller details has progressed. There's also a fundamentl assumption that there are general rules to be discovered. So far, this assumption appears to be reasonable -- as we refine our theories, they generally seem to work better wherever we look (of course, if they didn't we would discard those refinements).
A: OK, so it's in principle possible there's a giant projector that surrounds the Milky Way that (via some laws of physics we don't understand, so we have no way of telling that it's faking it) sends us light signals, etc. so our observations of the stuff "outside our galaxy" is actually from the projector. And then outside the projector, you actually have repulsive gravity, 128-dimensional spacetime and functional governments.
This is technically possible, and it's meaningful too, in the sense that you can fly to the projector region and check. It's also possible that this projector exists right outside the Earth, since I've never gone out and checked, or my house, and my memory of everything outside of it is implanted. Similarly, it's also possible the laws of physics will change on June 30, 2018 (if you're reading this after -- how do you know they already haven't?).
But the point is that this is all simply really unlikely. The (idealised) way of doing physics is to assign prior probabilities to each theory based on how complicated it is (this can be measured precisely, via Kolmogorov complexity), then look at the experimental data and apply Bayes's theorem to see how it affects your probability distribution. This is called "Solmonoff's theory of inductive inference", although I prefer to call it "Solmonoff's theory of inductive interference", in the sense of your data interfering with your probability distribution in some way.
Since having physical laws that change based on where you are is a theory with extremely high Kolmogorov complexity, you'll need very strong evidence to change the Bayesian confidence significantly, i.e. extraordinary claims require extraordinary evidence.
A: The power-tool of science is abduction.  No, not the kidnapping of a person.  Abduction is an inference mechanism, similar in nature to deduction and induction.  It's a way we infer truth.  Abduction is the inference that the best hypothesis is true.
We use this all the time, in science and in life.  If I toss a ball at you, it is highly unlikely that I'm using some rigorous approach such as Bayesian inference to determine the likelihood that the ball I'm throwing at you is not actually a nuclear detonator set to destroy the world if you fail to catch it.  That possibility doesn't even reach my mind, except when writing pedantic stack exchange answers.  Instead, I infer that the ball is actually a ball, and that my understanding of physics is actually correct enough to throw the ball at you without ending the world.  I used abduction to reduce any number of absurd possibilities down to just "the best" hypothesis, and acted as though it was true.  Occam's Razor is one example of an abductive inference method.  There are many others.
Abduction is tricky.  The term "best" has all sorts of nuances (as you can read about in the SEP article I linked above).  It can even catastrophically fail.  However, it is such a useful inference mechanism that we humans use it all the time in every day life.
Several of the other answers argue that we don't "think" that the laws of physics apply everywhere.  They wisely point out that science actually produces models which are not-provably-inconsistent with our observations.  These are both local observations of local effects and local observations of far away effects (such as looking through a telescope).  This is technically the correct answer.  Science never tells you the truth about anything.  Ever.  Nor does it ever claim to tell you the truth about anything.  That is the pathologically pedantic truth about science.
If you think science ever tells you the truth about anything, that should indicate that you expect people to use abduction to infer statements about the world to be true from the scientific models.  Thus, if one expects to hear "the Higgs Boson is real," based on evidence suggesting that we are 99.999999999% certain that CERN has detected them, then you expect that person to engage in abduction.  Which is natural.  Real humans use it all the time, and scientists are human.
Why they do not teach the concept of abduction in science class, I will never understand.  It is the cornerstone of applied science.
A: tl;dr-  We don't actually believe that the laws of physics are perfectly accurate, precise, or immutable.  Instead, we tend to work from the observation that the universe seems consistent with certain models as far as we can tell.

We haven't gotten to explore distant galaxies yet.  And given that the observable universe isn't 200-billion light-years wide – it's less than half that in diameter – we really don't have much to work from.
Example:  We don't believe that the speed of light is constant
For an extreme example, we often say that the speed of light, $c,$ is a constant – however, scientists don't believe it in an absolute sense.  What we actually believe is that the speed of light seems consistent with a constant so far as we have been able to tell.
If we did take the speed of light being constant to be literally and absolutely true, it'd imply stuff about how smooth spacetime must be and answer structural questions about scales at-and-below the Planck length.  Unfortunately, science isn't that easy; 'til we can meaningfully test how fast light moves between two points ${10}^{-100}\,\mathrm{m}$ apart, if such a test is even physically sensible, we can't claim light to move at a constant rate at that scale.
The speed of light is an extreme example since its constancy is such a cornerstone of modern physics.  The point's that we don't generally assume even the most cherished scientific assertions to be absolute; it's all about accepting the apparent consistency of an explanation in its correspondence to observation until we have a new explanation that corresponds better, applies more widely, is easier to work with, or/and has some other merit that makes it worthwhile.
We don't believe the laws of physics are the same in other galaxies
We don't believe that the known laws of physics behave exactly the same way in other galaxies.  Instead, what we've got are a bunch of models that seem to work better than any known alternative in the contexts in which we've tried to develop them.  So if we must speculate about how things work in a far-flung context, the best we can really do is tentatively extrapolate until experimental verification can provide us with more insight.
So, maybe the fine-structure constant, $\alpha$ varies over the universe; perhaps we'd one day describe some sort of universe-scale physics that causes it to vary.  But, 'til we have some mechanism to describe it, what can we really do?
Historical analog:  Atomic physics
In the early 1900's, scientists were working with trying to model the atom.  Their early attempts were largely based in the physics that they already knew from human-scale physics, e.g. the Rutherford model and Bohr model for atoms.  They basically tried to force observations into the framework that they already knew, then relaxed the framework as that didn't quite work.
Exploration of the distant universe may work out similarly.  This is, we'd likely try to fit everything into the models that we've got, then relax them as necessary to capture observations that we can't make fit into existing models.
Of course, this doesn't mean that we believe or disbelieve in our current models applying.  It's just that, until we have cause to suspect otherwise, we tend to suspect that our current models are more likely to be useful than models that we have no reason to suspect to be useful, e.g. random speculation.
A: While the answer above seems to have covered most important points, there is something that I'd like to add.
Things like the laws of gravity, the laws of momentum, and the laws of thermodynamics are built into the fabric of the universe itself - they are not just temporary rules that happen to be enforced right here in our neck of the woods.
With regard to how far away these galaxies are, and how long ago they existed: as far as we can tell these laws are not only built into the universe everywhere, they are built into the universe at all times as well.  Scientists universally accept that they are much like the rules of addition: 2+2 equals 4.  This is true not only here on Earth now, but everywhere in the universe at all times.
I suppose someone with a really vivid imagination could coin a question such as "how do they know that 2+2 doesn't equal 5 somewhere else in the universe, billions of years ago?"  I suppose in the strictest sense we really don't know for sure what 2+2 added up to billions of years ago, but science doesn't make progress by playing doubting Thomas at this level.
Lastly, I share your frustration about our inability to predict the weather far in advance.  However our limitations about weather forecasting are of a very different nature than the galaxy conundrum that you describe.
A: One interesting result not yet discussed is the fact that Noether's theorem mathematically links conservation of linear momentum and the invariance of laws of physics with respect to spatial translation.
Basically, if we observe that linear momentum is conserved then it is a necessary conclusion that the laws of physics are the same regardless of spatial location. This certainly doesn't cover all laws of physics, but only laws related to the fundamentals of motion, but it's still an interesting result: a local experiment allows us to make a deduction about the entire universe.
P.S. Noether's theorem is usually formulated by showing that a symmetry implies a conservation law and not vice versa, but I believe that the inverse does apply to conservation of linear momentum.
A: I think the solution to this may be to check out Occam's razor. That leads to the idea that we accept the simplest theory which matches best with what we observe. If you're asking why we don't believe that the spacetime relationship changes drastically (among other claims), it's because:


*

*We have no reason to believe that's the case. There's no evidence which needs to be explained by such a model. No observations or reasoning suggests that other galaxies are governed by drastically different laws. 

*We like symmetry. We have evidence that things work in a certain way around Earth and the observable universe, and hence are compelled to believe that the same laws are applicable at all scales, until we have reasons to believe otherwise. String theory predicts other situations, but those weren't observed yet in reality, and they don't emerge from a crude "Hey, why not?!" speculation.
That being said, though we believe that the same laws apply, we know that there are different physical phenomena going on in other galaxies. For example, this link will show you that there are different types of galaxies which behave differently in spite of the same laws, because of different initial conditions.
And to answer your reference to weather, that's chaos theory, and it deals with the dependence of the weather on extremely small factors which can't be observed reasonably. Check out the work of Edward Lorenz (http://eaps4.mit.edu/research/Lorenz/publications.htm). A gist of one of his most important experiments is that he ran the same weather simulator algorithm twice and got two entirely different predictions., even though he only neglected the 5th or 6th decimal place in one of input datasets. The initial conditions were different in such a minute way, but the simulation algorithm yielded incredibly different results! That doesn't seem particularly relevant to whether (no pun) or not there's a symmetry of physical 
laws. We know there's a huge number of factors in the prediction of weather, so our errors are huge. But our attempts to observe what's going on at other scales and locations are relatively error-free.
In one sentence: it's easy to believe in the symmetry of laws, and there's no reason yet to doubt their accuracy.
A: 
What if, say at 135 billion light years away, all of a sudden the time
  space relationship changes drastically

An interesting idea.  Do you have any reason or evidence whatever to indicate that this is the case?  
Let's play with the idea a bit.  If such a boundary was seen, additional questions would include whether the boundary was:


*

*Approaching, retreating, stationary (in what reference frame)?

*The mechanism of such a boundary?

*Energy considerations:  when e.g. a star or planetoid crossed such a boundary, how would its energy level change?  Explain where the (presumably enormous) energy would come from, or go to, for transitions either way across the boundary.


Do you begin to see the level of complexity that this would add to models of physics?  It seems to me that physicists would need to see something more than "what if?" before accepting such an idea.  I do expect that scans have been done to see whether there are such anomalies, and that we all would have heard about it if they were found.
As other posters have pointed out, the evidence we do have shows no such discontinuities.  Spectra of distant luminous objects are consistent with those of nearby objects, allowing for red- or blue-shift.  
Unfortunately there's a shortage of warp drive ships, so we can't go see just yet...

Lets be serious, we can't predict the weather farther out than 10 days accurately, and usually not that long....

Don't confuse changes to fundamental physics with our ability to make detailed predictions about an extremely complex, changing, chaotic system (a spinning Earth with a complex topology, churning layer of atmosphere, oceans with currents, lakes and streams, evaporation, condensation, insolation from the Sun, etc.) given a very limited number of data points.  
In the case of weather prediction, if we apply a hypothetical "what if the fundamental physics changed" then that might translate into, oh, things like:


*

*Gravity shifts

*Matter disappearing / appearing (excluding radioactive decay)

*Energy disappearing / appearing (excluding radioactive decay)


which, to my knowledge, has not been observed.
