For the carpet, is it the carpet in the bunny room or the carpet in the rest of the house when you let him out to play? If it’s the bunny room, you could place something over the carpet – cheap linoleum, vinyl carpet protector, etc and NIC cube off the edges so he can’t chew the flooring. For out and about, a spray bottle of water is the best way to stop it.
For the bed… what I did was since our bed frame is wood, we got a piece of untreated wood paneling and had it cut to size at home depot, then tacked it up to the frame, effectively turning it into a box bed and the one side that is open is against the wall. Another thing you could do is get cardboard boxes and weight them so he can’t move them and put them under your bed so he can’t get in there. Or, attach the x-pen to the bed frame with zip ties so he can’t move it.
For the doors, Powder is the same way, so instead of even closing doors I just baby gate off closets and stuff, it’s more secure than a door that doesn’t close as securely as the front/back doors do.
The humping uh… can’t help there, but with dogs sometimes a smell or something can set them off, maybe something you have on you is “exciting” him?