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.
Assuming open flame heaters are allowed in a garage in your area, carefully check all clearance requirements for the stove and flue. I would encourage you to adapt to triple wall insulated flue pipe at the ceiling of the first floor and stay with insulated pipe the rest of the way. Do not attempt to change back to single wall pipe.
You may want to install a couple of floor registers to help take some of the warm air from the first floor to the second floor. Remember, heat rises, and depending on how long you run the stove, the heat will eventually fill the second floor. Some simple electric fans can also be placed strategically to move the warm air around and even out the temp in the building.
Best Answer
That will be up to the LAHJ - or Local Authority Having Jursidiction. Figure out who that is (if you need building permits, it's usually whoever issues and/or does inspections related to them) and ask.
Don't be overly shocked if the answer is no - I suspect most RMHs in dwellings are located in places with no or very minimal inspections/code enforcement, as the building code folks tend to be a bit conservative about fire and houses - not having people killed in house fires is a big part of why they exist. As such they can be very slow to accept non-traditional approaches to containing fires in houses. On the other hand, they may be able to suggest an approach that keeps them happy.