How do we observe the expansion of the universe? This is my first question.
So I was reading about expansion of the universe and from what I've seen, the only way that we know that universe is expanding is by measuring redshift.
Also, here it says that energy is lost both in photons and other particles.
So my understanding is that we know that universe is expanding by measuring energy lost in light coming from distant stars.
Is there any other proof except from that?
Is it possible that energy is lost due to something else?
 A: There's an old theory called "tired light" where the momentum is lost due to waves hands some other reason, but as far as I'm aware this has been pretty much discounted these days.
The background behind the current-best-theory is this:
When you look at light from a star it's not a smooth spectrum, it has a series of dark lines in it, an "emission spectrum". These lines appear in very specific positions depending on the atoms that spat out the photons - Hydrogen, for example, produces a "fingerprint" of lines that are unique and can easily be identified.  (You can make your own spectrometer to view these lines using a CD and a cereal packet, lots of guides if you google it, it's a cool little project, just don't go staring directly at the sun!)
These lines, however, don't appear exactly where they should when you look at light from distant galaxies, they're moved a little to the red end of the spectrum, and this happens for all the lines - Hydrogen, Helium, Carbon etc.
This happens because the star/s producing the light are moving away from us and we observe a Doppler shift, the classic analogy being the way that a police siren changes pitch as it's driving away from you.
Hubble discovered this "red-shift" (frequently referred to as the "Z value" or just "Z") obeys a linear relationship with distance, suggesting the entire universe is expanding at a steady rate.  The distance, incidentally, is found using a number of different methods, such as "standard candles", objects which emit a known amount of lights, so we can work out distance by measuring how much fainter it is.  Redshift doesn't directly measure distance, it measures relative velocity, but once the relationship between redshift and distance was discovered and "calibrated" by using standard candles etc then it became a "proxy" for distance, a good rough-and-ready estimate for observations that had no other easy way to measure distance directly.
Later work showed that this isn't 100% correct, the expansion is actually accelerating (the "Dark Energy" problem), but the general interpretation of red-shift as a sign of an expanding universe is still the one that fits all of the data the best. 
Note that some objects are blue-shifted - the conventional explanation being that they are moving towards us, and any alternative theory is going to have to explain how and why some light gains energy instead of losing it.
A: As far as I understand, it has something to do with Hubble's Law.
Essentially, based on looking back at energy density of a distant star at one point, and then looking at it again, and determining that it has diminished, or something along those lines.
I assume it's a far fancier version of looking at a light you just passed as you drive down the road. If after a few seconds you look again, you will note that the light seems dimmer. If you really want, I am sure someone could provide you with a fancy formula for all that. 
