Should I drywall and/or insulate my garage wall?
I would insulate it and drywall it. 

You are already losing some heat from the house walls and the second floor into the garage. If you insulate it, your garage will be warmer - at least, after the door has been closed for a while - and you will lose less heat. It won't be super-warm, but it will help. 

You do not need 5/8" drywall for the garage walls. In the US, you need it between the garage and living space, but not on the exterior walls. 

I also recommend thinking about adding some outlets before you close the walls up - so much easier than doing it later.