I think there are some places designed specifcially for this. They're called "extended stay" apartments or something. I think, I'm not sure. The ones I've seen are not located in cities.

yes, you can! I did that with my boyfriend in New York. We rented an apartment for a month. It was airbnb.com we used. All vacation rentals.. It is reliable. Though I suggest you to view the place in advance, if you have the chance.