This is certainly not the response I get when I engage autopilot. I always make sure everything is stable before engaging AP, and that I have several hundred feet of clearance to the terrain, as the nose does dip slightly, but then recovers immediately.
A few more details would be useful. What plane are you in? At what point does yours go nose up, eg when you first switch on AP, or when you engage ALT etc?
The banking to the right does sound like a hardware problem, as it’s in every aircraft. What peripherals are you using?