Is my HVAC system unbalanced?
A whistling sound at the return often indicates the grill is too small relative to the duct attached to it.

Does consistently leaving the doors open allow the system to sufficiently heat and cool the home?

For most bedrooms with more than 100 cfm of supply aor a 1 inch door cut is not sufficient opening for return air. If you have carpet installed it is likely you have less than 1 inch. Transfer or jumper ducts may help alleviate the pressure the home inspector noted. This can be easily checked with a manometer. The pressure difference between the room and the central area where the return is located should be 3 pa. or less.

Experiment by leaving all the doors open (as much a possible) for several days first. If leaving the doors open and/or providing better return circulation does not resolve the issue, then other help is needed.

Based on your other it sounds like you need a competent HVAC tech to troubleshoot and balance your delivery system. It is also possible that you are undersized or there are some other issues (like excessive duct leakage, or set up issues) if you are having trouble maintaining the desired thermostat set point.