Resolved issue 1 by replacing the battery and charger. Issue 2 still happens but I may be getting a newer used laptop later this month.
Unfortunately, I am dealing with stripped screws and so do not have access to the interior.
I have dealt with the thermals somewhat by throttling the CPU to 25-50% and keeping the fans on; esp. useful during hot summer days when my room is 15C hotter because of the laptop. I wonder how many people actually throttle their CPU. Most seem to want to overclock. 