that valve thing is actually pretty funny.
it is a 4 seasons valve inside the box.
I too had an issue where I was getting warm air after the car warmed up.
they said I had the lines routed wrong.
it needed the initial pressure loss from going through the heater core to allow it to seal.
routing it through the heater core first fixed it.

I replaced the valve with a new one a month ago, thinking it was sticky but turns out the electric motor can't hack it anymore.