We know that some galaxies are moving away from us faster than the speed of light and we know it by measuring the redshift, but how's that possible? If they're moving away say at 2c, how would the light of the galaxy even reach us? How do we measure "redshift" for something faster than light?
The following papers give good explanations:
In summary, Hubble Law: $v = H(t)D$, where $v$ is recession velocity, $D$ is distance, and $H(t)$ is the Hubble "constant" at a given time, requires that beyond a certain distance velocity is greater than the speed of light. If recession velocity at the location of a traveling photon were greater than the speed of light the entire time the photon from a distance galaxy were traveling, we would never observe the photon. A photon emitted from a galaxy moving away from us faster than light, initially is also receding from us. However, the photon may eventually get to a region of spacetime where recession from us is $<c$. In this case, the photon can reach us. The exact relationship between red shift and velocity depends upon the cosmological model, but according to the above references, galaxies with red shifts greater than ~3 were and are receding from us faster than light.
Only if the photons from the galaxy reach a region of spacetime where recession velocity is $<c$.
Red-shift is measured as the change in wavelength of the light, but rather than interpreting the results using special relativity (which would result in $v<c$ for all red shifts), the results are interpreted in the context of a cosmological model and general relativity.
Light from beyond the Hubble sphere (the place where recession velocity equals the speed of light) reaches us daily.
I'm not good enough a physicist to come up with a nice layman's explanation for this fact, but it might help to think in comoving coordinates: This is a special coordinate system where the coordinate grid expands with space, ie even though the proper distance between galaxies will increase, their coordinates won't change.
In this coordinate system, light does not get frozen at the Hubble sphere (as one might possibly expect), but steadily moves from emitter to eventual observer, regardless of any change in proper distance.
The moving steadily towards us should actually also hold true for light emitted from beyond the cosmic event horizon (the thing that actually delimits the observational universe) - it just takes the light a longer-than-infinite time to reach us ;)
As to the second part of your question about the redshift: That doesn't depend on recession velocities, but rather on relative velocities as computed by parallel transport along the light path (and should stay below $c$ until you hit the event horizon).
I do not know if the following answer can explain each and every observation, but here goes :
The expansion or moving away of galaxies is dependent on the distance between them, if something is moving away at some rate then previously since it must have been close, it must have moved away at a slower pace.
While making astronomical observations, we are always seeing back in time. 8 minutes back to see the moon and millions of years to see some far of stars. The light of such stars that would be leaving the star when the observation is made will reach us by the time star already dies, it is interesting that the star much before that may have reached some place from where it moves away at speed larger than that of light, and hence it can no more be observed.
Astronomers after observations, calculate the present state of the cellestial bodies and then publish all results, so if they tell you that something is a million light years away and is moving away at 2c, then that is its present position and speed, it was observed due to the light it emitted long tims ago and various observations and calculations allow us to predict its present state.
I am no specialist in gravity or cosmology. Though, I know (without details) that A. Peres proved that the light velocity wasn't the same all along the history of the universe. The reference is
Int. J. Mod. Phys. D, 12, 1751 (2003). DOI: 10.1142/S0218271803004043
International Journal of Modern Physics D (Gravitation; Astrophysics and Cosmology)
Volume 12, Issue 09, October 2003
VARIABILITY OF FUNDAMENTAL CONSTANTS
ASHER PERES, This essay received an "honorable mention" in the 2003 Essay Competition of the Gravity Research Foundation.