A/C compressor engages when A/C is not on

Do you notice the A/C running when in defrost? Or is it in every mode? If while in defrost mode, it is normal. The system runs in defrost to dry out the air in an attempt to keep the windows from fogging inside. If it is running in other modes as well then you have a problem. It may be related to having no heat if the controller thinks it is always set to A/C. That would tend to run cold air through the heater core. Do you hear any change in airflow when you switch from hot to cold and back?

Yeah, that was sort of snuck in there because they changed the way the systems worked because of the venting systems changing over time.
If the in line is hot and the out isn't it is most likely the core is plugged. You can try to reverse flush it using a garden hose or a flush gun. 50/50 as to it working as it depends on what is plugging the core.
Lot's of videos out there on flushing. However, if the core is failing unplugging it could lead to leaks.

If you're going in for the core, replace both the heater and AC cores and do the upgrades for the doors and actuators while you're in there. There are also a few spots that you can add sound deadener pads to prevent any dash squeaks. The service information we have shows how to change them out but a video is a lot better.