1. Why would the solenoid work in one direction, i.e., close, but not open, with 6 volts?
I would guess that it would close without any voltage but not being familiar with the unit I can’ say. I have a gas heater that shuts off the gas during any power failure.
2. Why would increasing the voltage to 7.5 volts allow it to open and close, when it is designed to open and close with 6 volts?
It could be due to an increased resistance in the winding
3. Does a solenoid wear out slowly, or does it go all at once?
It can go either way, slowly or all at once
3. Is there a way to add an additional battery to the receiver, so I don't have to mess with trying to replace the solenoid, if the solenoid is defective?
I am not sure that this would be a good idea