Air conditioning

You may need to flush heater core? The water pump slows down at idle, speeds up at higher rpms. Coolant flow through heater core slows down at idle. If heater core is restricted, even less flow at idle.

If the blower only worked only on high, I'd think there was something wrong with the resistor. I believe with the speed on high, the voltage bypasses the resistor and goes to the relay, then to the blower motor.

If you can get at the connector for the blower motor, you can take the connector loose, then run B+ and ground to the blower motor, to see if it will run, if it runs, you know the motor is ok. You should use an inline fuse on the voltage circuit to the motor.
I don't have any wiring colors, but, black is usually ground.

Why did you clean and flush lines to begin with? Was their a problem to cause this? And what are your pressures? Not enough info to give a good opinion

Easiest way is to just swap it out with a known good relay that has the same pin placement. or you can check out our tutorial on testing a relay at https://www.2carpros.com/how_to/how_do_i_check_a_relay.htm

Sounds like the mixing doors are somehow stuck could be a vacuum door or electronically controlled door, you wont be able to tell untill you inspect it.