In a woodstove, where the fire doesn't get very hot, you get a lot of unburnt gases. These are compounds that could burn but only at a higher temperature. Instead, they evaporate. As they travel up the chimney, the cool and condense. Over time it builds up. If you then burn a really hot fire, they can finally ignite, and your chimney burns. This can happen when you use a lot of pitchy softwood for a long time (Douglas Fir, for example) and then burn some really dry hardwood (e.g., maple).
If you do something clever to extract more heat from the chimney, you will get more condensed goop in your chimney, so be sure to clean it regularly.
Another way to deal with this problem is to always make a really hot fire. Then almost everything will burn, leaving very little goop in the chimney, and very little ash. That's going to be too much heat for your house, so you have to surround your fire with an enormous thermal mass. That will moderate the temperature. This is the principle behind a "Russian stove", among other names.
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.
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As one comment mentions, the type of old stove you're describing heats by radiating, and it will be difficult to move that heat into the house somehow.
Another issue to be concerned with is whether or not the stove is certified - if it's not, chances are you won't be able to get insurance, or insurance may be invalidated if it's the cause of a fire. Check your policy.
There are some good solutions for an outdoor wood furnace, but they're not going to be cheap. It's far easier and cheaper to install a good quality, modern wood stove in the basement, and it's (IMHO) the nicest quality heat you'll get.
When we moved out of a house that had a wood furnace (which heated via forced air, and also had a boiler to heat water for the baseboard radiators) into a house with electric baseboard, the first thing we did was install a wood stove.