Hey Sam -
We checked out all of those buildings and ended up at 77H, which we're excited about. It far exceeded everything else we looked at, especially the size and finishes of the apts. and it was competitively priced. It's brand new and the building and amenities are great. It is literally a three minute walk from campus. Because they're brand new and working to fill their units, we were able to sign a lease on the spot (end of March), with an incentive, and we're not moving until mid-June. I can't recommend the building enough. According to the staff, it is filling up quickly with GULC students.
Senate Square and 360 H are next to each other. We weren't thrilled with the location - they're also on H street, but further away from campus than 77H and in the midst of a part of H street that is still being developed. There are a lot of restaurant options nearby and it's up and coming, but still looks/seems a bit dilapidated (and we didn't find the buildings/apartments/staff as nice as 77H).
Mass Court was decent and is probably closest to campus other than 77H (in a different direction). The apartments were standard, very dorm-like...hip hop was playing in the lobby and there was a lot of action (mostly students) in the common areas. The staff was nice and it was reasonably priced.
We checked out every building within walking distance of campus, so I'm happy to answer any questions or share our photos, observations, etc.