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what controls the taco zone control?

that t-t is designed to control a boiler when the zone is calling, it closes t-t on a heat call or a valve end switch depending on how it is set up.

Doesn't the air handler have a transformer and relay to control the blower? G and Y and Rc from the t-stat should connect to the air handler and the contactor in the condenser should connect to Y and C in the air handler.

Or do you have a hydroair coil in the air handler? Old systems g on the t-stat called the blower on high which is what you need for ac and y was just a junction point to connect the t-stat and condenser contactor. In newer systems y turns the blower on high, so you'd need a second relay for y if you don't have a control in the air handler to do a high blower call with g or y.

Every part in that control box is easily replaceable and a standard part.

What is what you've labeled T-T? normally it is a control to a boiler that turns the boiler on when you short the terminals. Is it controlling a relay that shorts t-t in your case?(you need to make sure this is correct)

the easiest way to add cooling to what you have is probably to add a second relay that is controlled by y from the t-stat, use that relay to power the cooling speed of the blower, and connect the condenser contactor between c and y at the air handler.

What happened? Did this system ever work before? Or did you just buy this place?

I would also prefer to put ductwork in. The home will be much more comfortable. Mini splits need to be installed on an exterior wall so you can drain the condensate outside and access the line set fittings from the outside. This limits where you can install the indoor units.

I know you technically can install them on an interior wall, but anytime that is done it becomes a nightmare for service. Condensate drainage usually requires a pump which can be problematic. Personally I think installing them on an interior wall is idiotic.

If you can install a proper ducted system you will be much better off in the long run.

Wall mounts are not zero maintaince so you want to minimize the number. Single head per floor can work if you can get the building load down enough, possible as part of the reno but takes some planning. You would still need a bit of supplement heat in the rooms so you have heat there overnight with the doors closed.

A better option is what @Larry Weingarten suggested.

Steam boiler generally means basement. That means you can put the main floor unit there and run the ducts in the basement. If you want to ever finish the basement, try to put the supply trunk against an outside wall so it doesn't become a headbanger. The small bed on the main might need some baseboard heat as it looks like there is no basement bellow it.

The 2nd floor unit can also be in the basement with a trunk up to the 2nd floor say with the trunk in the corner of the main floor living. You can also put it in a dropped ceiling of the main floor bath and have a small bulkhead run across the main floor for the trunk with supply takeoffs in the floor space.

In cold climate you want the supply registers under windows.