To answer your questions:
1) I live in Arlington and have jogged through the GWU area many times. The sorority townhouses look really cute, but I'm not sure they can house a lot of members.
You could see if you can board in a sorority house. Contact GWU about that policy.
Even if you do board in the sorority house, you would still have to go through rush.
FYI- boarding in sorority house XYZ does not automatically give you a bid to XYZ.You will still have to attend their rush events.
2) Epsilon Sigma Alpha, a national service sorority, has a chapter at GW. That's another option.
3)DC has good night life. Georgetown has great restaurants and bars. Dupont Circle is a nice walk from Foggy Bottom, also lots of fun. I haven't been to Adams Morgan or U St/Cardozo in years, but those areas used to be lots of fun. If you like punk or indie music, you can go to the Black Cat on U St. (U St was dangerous, so I didn't go that often.) Arlington, VA is a short metro ride from GW's campus. We have loads of restaurants and two music venues, Iota and Galaxy Hut.
Good luck and let us know how it all goes!