A/C compressor turning on and off when the heater is on?

Totally normal on modern vehicles. In any mode that uses the defroster outlets like defrost or recirculate or in a split mode it will cycle on/off in an attempt to dry the air entering the system. This is to keep from blowing moist air onto the glass and causing it to fog or freeze up.

Okay, if it is doing it with defrost off and recirculate isn't working can I ask what the outside temperature is and if this truck has the dual zone or automatic style of heater control?
With those the outside temperature can cause some changes but recirculate should still operate.

Being this isn't the automatic system where you set it to a specific temperature, I would start with a dealer level scan tool that can communicate with the HVAC module and see if there are any codes stored in it. It may be that the recirculate actuator isn't sending the correct signal to it or the module itself is failing. Using a scan tool should show if there is an issue. You don't hear any clicking when you select recirculate do you? That would be a sign of a failed actuator.

The blink means there is an issue with the actuator signal. That could be the actuator itself or the control head, however with the AC coming on at random as well I would lean more toward it being the control module. As a test to see if it is the module or the actuator you could do a reset on the system and see if it clears the issue. If it does it is likely the control module. If it still stays "broken" then it is more likely the actuator.

The process is simple to do, With the truck off remove the HVAC/ECAS fuse for about 20 seconds. Now put the fuse back in and start the truck. Do not touch any of the heater controls. After one minute the calibration procedure will be complete. Test the system and see if it's better. If it is OK now then it's likely the control head. It isn't hard to replace but I would test it first.
To replace it you remove the trim panel that goes around it, remove the screws that hold it in place and swap it out. Then it gets programmed using a dealer level scan tool.