Yes, Houston traffic is rough, but it probably won't be much different than what you already experience, maybe a little bit better if you plan to live in the Med Center (TMC). Most people live outside of the 610 loop, so rush hour traffic is worse on roads leading to and including 610 and 59.
Yes, $900 is on-par, and actually reasonable, for a 1bdrm apartment in the TMC. Anything cheaper and you risk being in an unsafe neighborhood. Also, parking in the TMC is terrible. It's expensive and very competitive. I did not go to RIce but I've known Rice students who biked. Check with the school about parking, they may have special parking/parking rates for grad students. Oh, and be sure to lock up your bike, people WILL steal it!
Rice Village has great pubs (O'Neill's and Gingerman to name a couple), coffee shops and eateries. It starts at the corner of Kirby and University blvd. There is also a fancy new HEB just north of there, a Randalls southwest of the village on Holcombe, and jogging/bike trail along the Bray's bayou that transverses the TMC. Houston has plenty of coffee shops and restaurants, you'd be sure to find a few you'll love.
Apartment openings are year-round, but that may be more competitive for apartments closer to rice. Medical and professional students swipe up apartments in the TMC before each semester so if you cannot visit until late summer, you might consider using an apartment locator. They can go a look at apartments for you, plus they are free to you and will have plenty of information about specific apartments and neighborhoods.