In order, and thank you for asking these questions, they’re super helpful!
Which one is the drain back the tank valve? I only have two valves I can open/ close - the one to the tank is open and the drain valve is closed.
Nope, no water or air come out of the faucet with the pump on.
Valve is open, but I’m not sure if the hot water tank is filling as no water/air releases when I turn the tap to hot with the pump running. No water or air release with the cold tap turned on and pump running, either.
To my knowledge, and please correct me if I’m wrong, the only way for water to exit the system is through the tap, shower, drain, or water heater drain plug. I have tried the tap (see results above) and the drain (which released a small bit of air the first time but not since). I have not tried the shower, though. I will do that tomorrow morning and report back. Should I even mess with the water heater drain plug?