As others have suggested, look at Columbia Heights or Shaw for places to live. Though a little further (just past Columbia Heights), you can find good deals in Petworth, too.

As for working as a messenger, get to work early (around 8am) and be prepared for a long day. The Summer is going to be slow, especially when Congress is out of session. Also, a lot of people try to get messenger jobs in the Summer, so work can be pretty scarce. You'll need to stay with a company for a while, and work year-round, to really get a decent amount of work/money (though if you're looking to make lots of money, messenger work probably isn't really the way...).