You should be just fine flying one handed. You will have to prioritize much more than other pilots but it is possible. It's is going to be quite demanding trying to land the airplane but it is doable. I once landed a 172 starting from downwind without touching the controls just trim, power and rudder pressure.

I doubt you will be able to get a medical from the AME. Not that big of a problem just have to fly with a current and qualified pilot in command.