Are there certain parts of the Universe that are expanding faster than others? Is there anyway to definitively measure this?
If dark energy varied by location, then plots of 1a supernova brightness vs redshift should vary depending on which direction you look in the sky. AFAIK, that's not the case. For example, although coords are not accounted for, there's not a lot of scatter in this plot:
In the relationship between the distance and redshift of Type 1a supernovae, the data (points) agree with the equation in which light propagates through the expanding universe on the least-time path (solid line). Image credit: Annila. ©2011 Royal Astronomical Society
Good question but we can only see out a certain distance because of it's expansion. AFAIK acceleration of the universe will eventually reduce the distance we can see over time.
No one knows if the universe is infinitely large, or even if ours is the only universe that exists. And other parts of the universe, very far away, might be quite different from the universe closer to home. The site http://skyserver.sdss.org/dr1/en/astro/universe/universe.asp give a good basic picture.