Bedroom not getting enough heat

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.

Variation of the temperature from room to room that is different from the thermostat is a basic fact of operational behavior. you will have to get used to some variation!!

It is possible to even out the differences by changing how much of the heated air gets to each room. This is usually done by using vent controls on each room to permit more or less air into each space. Sometimes there are also vent pipe dampers or diverters that can be adjusted as well. A balancing exercise is usually an iterative process that takes time and patience.

A well balanced system will only stay that way if the same conditions apply as when the balancing was done. Some of these conditions would include outside air temperature, state of window coverings, whether doors are open, closed or partly ajar and if windows remain in the same state (closed or open).

Note that a system balanced for heating can take an entirely different set of adjustments to optimize for cooling during warmer parts of the year.

So as you can see you could try making small gradual changes in your setup. Try closing off more of the the vents that feed into the area where the thermostat is located and open those in the colder room a bit if not already fully open. Remember - iterative process.