1. YES, the primary relay is closed(ON)when the key is ON.
2. Never checked before about how hot they get
3. That relay is ON with the Key, because Tiffin slides operate with the KEY ON. So, for the pump motor solenoid to activate from the slide switch, the primary solenoid must be ON to transmit battery power to the motor solenoid
4. The solenoids are not that expensive. I would replace it and then see.
5. Are you the original owner? Maybe someone replaced the continuous duty solenoid with an aftermarket that does not have the same rating as the HWH solenoid Doug
PS Tiffin rarely knows how any Non Tiffin appliance works
E MAIL HWH! You will get a response in less than 2 days.