Cleveland Heights is the suburb just east of the university where most people live. Some places in Cle Hts are within walking distance, most are not. If you really want to walk to school then finding a place in little Italy is the best bet. Id say 90% of grad students live in one of those two places unless they have a family and dont mind living a little farther away like in Shaker Heights to have their own house.
Dont know of any leasing companies. Id say the best bet is to come here for a weekend, get a paper (the plain dealer) and start looking around. Might also be able to find places on Cleveland.com.
If you have any more specific questions let me know, might be able to help.