Hot Water Issue

I'd suggest that you check the position of any bypass valves in your system. If the water is hot in the heater tank, you should get hot water out of a faucet, regardless of any air pocket in the heater tank. There are some newer heaters that have a mixing valve where the water is heated hotter and cold water is mixed into the hot water supply. You may also have air locked the heater tank by not opening a hot water valve when filling the tank. That is essentially what you are doing with your "after-heating" process. I don't like using the TPRV for anything. I've had too many problems with broken O-rings and poor seating.

Pump runs when line pressure drops below a certain pressure level. This can occur because of a leak in the system, pressure deviation in the heater tank or lines, the pump leaking pressure back to the supply side.