My rear wheel isn't properly spinning. How can I tell whats wrong? I'm just getting back into cycling after a long hiatus. My bike is a little old and worn and its got some issues. The biggest one is the rear wheel.

My rear wheel spins, but it seems to be off center. When I look at the wheel spin the distance between it and the brakes varies. Also when I squeeze the brakes the break pads don't reach the wheel. How do I tell which part of the bike is messed up? Some thoughts I had:


The wheel itself is bent. Possible since it was in accidents before. 
The wheel is attached to the bike at an angle. Possible since I did self repairs before and I'm not super experiences in bike repairs.
The wheel isn't fully 'screwed in' (not sure what the proper term is) to the bike frame. Unlikely since it doesn't seem to wobble at all.
The brakes are messed up and the closer-farther effect is just an optical illusion.


Also possibly relevant: 3 spokes on the wheel are broken.

How can I tell which of these possibilities is causing my issues, or if its a different issue? And if so, how do I fix it?
You should take it to your local bike shop and get the wheel dished and trued. It is probably a mix of both those issues. Depending on how bad it is you may need to replace the wheel entirely.