Heater Issues
#2
The A/C compressor cycling too often will not cause the heater to blow cold air. This is usually a sign of the A/C being low on freon. The heater blowing cold air is probably the blend door actuator being bad. It could also be a clogged heater core. With the engine a temp. see if both heater hoses are equally hot. If one is much hotter than the other then you have a clogged heater core. Most likely the blend door actuator.
#3
New heater core a year ago. Actuator is functioning properly. I tested heater core and coolant will free flow through it. I tested the temp in the vents in the front of the dash and it was about 60 degrees. I pulled the relay out for the A/C and the temp in said vents went up to 140 degrees. One heater hose was hot and the other was cold until I pulled the relay. I do know the A/C compressor will cycle some to help pull moister out of the cab in defrost mode but it cycles even in if it on the floor or out the front vents.. If the HVAC controls are bad will it think your trying to cool the cab down and cycle the compressor?