For central A/C, you'll need ducts and vents. Once you have the ducting in place anyway, you might as well use it for both your heating and cooling.
For a three floor house, you definitely want some kind of zoned system, with a thermostat on each floor. (I wish my 2-floor house was zoned.) You may want multiple cooling systems, or maybe one system with motorized dampers to control airflow to the various floors. Your HVAC contractor should be able to make a reccomendation of what will work best.
Whatever you do, do not skimp on the ductwork. Your HVAC contractor should run calculations on how big the ducts need to be. For many home builders, HVAC is a minor concern in the bulding plan. Make sure everything is sized correctly. Changing ductwork after the fact is very difficult. And if you don't have proper ducting, it won't matter how good your central heating using is.
For new construction, Geothermal is usually a good candidate. It can provide heating, cooling, and hot water with crazy efficiency. The installation cost is MUCH higher though. (There are tax credits that will offset some of the cost.) If you aren't up for geothermal, I would go for a traditional air-sourced heat pump for your heating and cooling needs. They are pretty efficient (at least in moderate climates). You are pretty much in the perfect lattitude for a heat pump.
However, I would not have the heat pump (air-sourced or geothermal) as your only heating system. Some kind of alternate (non-ducted) heat source would be a good idea. Heated floors, gas fireplace, etc. Air-sourced heat pumps lose their efficiency in very cold weather. And it can also be nice to have gentle air convection, vs a duct that is blowing in your face. You also have a backup system in case your central heat fails.
That said, ducted air is nice, because some regular circulation of air in all the rooms of the house avoids that "stagnant air" smell, as well as cutting down on dust, dander, etc.
Also, central air systems give you options for humidification and de-humification of the whole house. Look into that. It is usually a minor cost increase in the system, but can make a major difference in comfort.
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.
Best Answer
This looks like a vent pipe that would be hooked up to a ventilation fan in the bathroom. If the bathroom door was closed, you might get quite a bit of air loss out of that pipe, but when the door is open, there wouldn't be as much pressure to push the A/C out of that room.
Those ventilation fans are not supposed to vent into the attic because their supposed purpose is to remove humid air, but incorrect installations are pretty common.