Do the laws of physics work everywhere in the universe? Do the laws of physics change anywhere in the universe? Or will they change from place to place in the universe?
 A: The aim of physics theoretical research  is to formulate mathematically the "laws of physics" in such a form that the formulae will work anywhere in the observed/known  universe. We can test well our immediate neighbourhood, analyse signals coming from the beginning of the Big Bang and expect that the final Theory Of Everything (TOE) will be consistent through the whole. Physics is not there yet.
A: There's another question on this site about whether the laws of physics change over time. I think that the answers to that one (including mine) apply pretty much perfectly to this question about whether the laws change in space.
We expect the fundamental laws of physics to be the same throughout space. In fact, if we found that they were not, we would strongly expect that that meant that the laws we had discovered were not the fundamental ones.
It's very sensible to ask whether the laws as we currently understand them vary with respect to position. People do try to test these things experimentally from time to time. For instance, some experiments to test whether fundamental constants change with time are also sensitive variations in the fundamental constants with position.
Some cosmological theories, especially some of those that come under the heading of "multiverse" theories do allow for the possibility that the laws are different in different regions of space, although generally only on scales much larger than what we can observe. In general, in such theories, the truly fundamental laws are the same everywhere, but the way the evolution of the Universe played out in different regions is so different that the laws appear quite different. 
One way this can happen is by the mechanism of spontaneous symmetry breaking. When the Universe cooled down from very high temperatures, it probably underwent various transitions, more or less like phase transitions, in which an initially symmetric state turns into a less-symmetric state. In those transitions, there may be different ways that the final state can come out, and they may be quite dramatically different -- completely different sorts of particles may exist, for instance. There could be different regions of the Universe in which the symmetry breaking went different ways, in which case the "apparent" laws would be utterly different in different regions, but probably only on scales many orders of magnitude larger than what we can see.
A: One thing not mentioned is Noether's theorem. This theorem states that if there is a symmetry in the system, there is also a conserved quantity, and vice versa.
Some examples of the theorem in action:


*

*Rotational symmetry $\leftrightarrow$ angular momentum. If you set up an experiment, its behavior doesn't change depending on the direction you observe the experiment from. This symmetry is rotational symmetry, and its existence implies angular momentum is conserved.

*Time symmetry $\leftrightarrow$ energy. The results of experiments don't depend on when it is performed. This is time symmetry, and implies energy is conserved.

*Translational symmetry $\leftrightarrow$ linear momentum. If the laws of physics don't depend on where the experiment is performed, then linear momentum is conserved.


All these things can be tested. If we measure conservation of angular momentum, then we know that rotational symmetry exists. Similarly, if we measure conservation of linear momentum - and we do - then the laws of physics do not depend on position. In other words, they do work everywhere in the universe.
