Can graduate or medical students stay in dorms?

This depends on the school. Some school don't have the dorm but normally are close neighborhoods close that they offer rooms with meal include, there is a website that you can look is call www.craigslist.org or add in the same school people sharing apartments, or local newspaper.

Yes, some schools have graduate level dorm or off campus housing.

Typically yes, but they tend to be for International students who are looking for a place to live while they get their bearings in a new place. Usually they will stay for a short time and then find somewhere else. Most grad students prefer to live on their own--you probably will too by the time you're in grad school. The only time it's an issue is in high cost cities, like Cambridge, New York, Berkeley, etc. Then you may find yourself needing to room with someone.