I rent also, and I have to say my landlord is very easy going. If I want to paint the apartment, I can paint it any color I want. He said I can plant anything anywhere in the yard, small as it is. I can make the whole yard into a garden if I want.
One thing I would LIKE to do next spring, is see if the zoning and my landlord would let me have a few chickens. I would like to turn my shed into a chicken coop. Or maybe part of it.
Another thing I think is a good idea if you have close neighbors or live in apartment buildings, get to know your neighbor and ride share when you can to the store or wherever.