How can I have door latch hardware at different heights on each side?
(Note that this door location isn't up to code. I'm assuming your house is rather old, but if it's not, it's indicative of other problems.)

Do you need the door to actually latch? If not, you could get 1 "ball catch" or "roller catch", and 2 dummy door knobs. Instead of actually turning the doorknob, you just push/pull the door open. The catch keeps the door closed until it's forced open.