Any ideas on why my "restore" key does not do anything?

Good question! It might be because of the keyboard contacts, a broken wire, broken trace, or bad I/O controller. I would take apart your C64 for further inspection and then come back to the question.

(Note: I am not a professional by any means and may be totally wrong. Also, I don't know the reason to problem two.)

This is easy to answer and is SUPER Common.
To fix this you have to replace the original capacitor of 51 pF is exchanged by a 4.7 nF one at C38.

There are a few good YouTube videos about this.