I'm not familiar with Columbia, but my friend lives in the Greenbelt area and loves it.
Also yes, finding studios in D.C. under $1400ish will be very hard, if they're around $1100 or below, they will usually have income limits, which might not be an issue on a grad stipend. A group house room around $1000 is much more do-able, but location dependent. I always plug biking in D.C. to get around the city and immediate suburbs! The infrastructure is always getting better, and the biking community is super awesome and active.
Lastly, I know this is potentially creepy, but my group house has rooms opening up late spring! And my room will be open starting August! We're in Petworth (residential, a little far out from downtown, but great community), a 10 minute walk from the Green/Yellow line, rents around $800, 6 people in total but spread across three floors and three bathrooms. One roommate has done grad school and UMD and Georgetown, another did undergrad at GW. Feel free to PM me if you're interested or have D.C. questions in general.