Temp gauge going to hot with only he key on not running and motor is cold.
This is the fix.
Temp gauge, Unhook it from the sender on the intake, with the key on It is the red wire with white stripe. The gauge should go to cold.
If this works you need a new sending unit for it.
The brake Light and turn signal not working. That is coming from your turn signal wiring in the steering colum. Replace the turn signal wiring.