Does my compost bin need to have an open bottom so the compost will be exposed to the ground?

There are versions of both types in use everywhere, and there are arguments for both. It's simply a matter of choice. You have to watch your carbon/nitrogen ratio a little closer with a closed bin to keep odor down but that is the only disadvantage I know of. A trash can with a bottom would certainly be easier to tumble than trying to turn the compost inside by hand. All you'd need is a few air holes in it.