Are foam brushes suitable for applying wood stain, or is a cloth necessary? Is "rubbing in" required?
Most stains don't need to be "rubbed in" but of course that depends from can to can. Read what your can says for how to apply. If it doesn't mention rubbing in then don't do it.  Almost all stain (unless otherwise stated on the can) has to be removed shortly after it has been applied. The longer you leave it on for before you wipe it off, the darker it will be. Try to be consistent on your floors for whatever time you choose to keep the color consistent across the room.

Using a cloth or foam pad doesn't really matter for the applying part. Of course it will be much easier to remove the stain with a cloth. I personally like using Viva brand paper towels as they are very rugged for this but you can choose whatever works for you.

As an aside, if you are staining floors, you may want to look into water popping as I feel it produces a much better end product. :)