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.
This isn't a DIY project. You need to have someone who has experience engineering these systems give you input. Steam radiator systems are not something that a DIYer should be designing; there are a lot of design aspects that are not intuitive and need an expert's hand.
On top of that, the question that you're asking here is one that you need to take up with the manufacturer of the floor heating system that you're considering and the manufacturer of the thermal storage and furnace system.
Again, this is not a DIY project -- it's not something where you can head on off to your local hardware store and they'll have all of these products in stock. You need to at least consult with all of the manufacturers of the components that you're considering, or you need to have an experienced firm design the entire system.
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Best Answer
It is an automatic air vent that is supposed to allow air to escape from the heating pipes. It should not be leaking water. From the looks of it, it is definitely defective and should be replaced. I recommend a plumber.