Actually, it sound like a problem that may be unique to you or your PC.
For me, the Default C172 steam works fine … Trim works as expected, as well as Joystick, and all Mouse over area in the cockpit (provide you in the appropriate view)
The comment “Also get a random high pitch beeping (stall maybe) when trying to move the yoke” may indicate that it may be “Pilot Error”
Check airspeed ( Greater than 60 knots ), and position of trim wheel (Trim wheel should be “near” center & NOT be at an End Stop)
If AP is on, turn it off, and get the Airspeed up, and the trim near to center, and then the plane should fly in a controllable manner)
Make sure MIXTURE is rich (all the way in) and Throttle FULL (all the way out) & altitude is low ( 1000 - 3000ft when testing.)