Winters are cold, but if you're from New York, you'll probably be fine. For safety, I would make sure that there is a basement. It doesn't necessarily have to be yours. When my husband and I lived together, we lived in a small little 2-bedroom apartment that was right behind campus (off 9th and Mississippi) and we spent around 600 for rent (this was 2006-2007). We were on the second floor, but if there was a tornado warning, we could go downstairs to the communal laundry room area. If you're looking to rent a house, rather than an apartment, most probably have basements anyway, but it would be good to have one. Tornado watches and warnings are pretty common, but I've lived in Kansas for 30 years and have never seen a tornado (besides people's amateur footage that gets sent to the news station). I've been hustled into basements while out doing stuff (including seeing Brand New at a show at Liberty Hall), but have never been in an actual tornado. So there definitely are tornadoes, but being in the wrong place at the wrong time to be in one is less common. I wouldn't be too concerned about them.
As for when you should start looking, you could probably start looking around now, but many leases go through July 31st with people subletting their places over the summer if they go home for the summer or graduated in May. We sublet a place one summer that we ended up staying in and renting when the school year started.
The fraternities and sororities are distributed throughout campus (Tennessee St., West Campus Road, and Indiana St. as they get close to campus). They might be something to avoid if you're needing more quiet. There are a couple of bars on 14th street right next to campus that (when I lived there) got pretty loud on the weekends and when students didn't have class the next day. So there are some areas to avoid, but you can easily be within walking or biking distance from campus if you don't have or want to bring a car with you.