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My question is why do u need a pump if u already have existing plumbing in your basement. Run the line to your existing plumbing.

6 gauge

If this is the window unit there is an internal heat sensor. I recommend u get a tech. They are not easy to repair.
This is not a thermostat problem.

It seems to me as if u have what u call zone valves. 2 small square boxes on the riser pipes to your boiler.
At the bottom of th silver the boxes there a little piece of metal sticking out.
The...

The fan will only work for a/c or fan only.
2 possible problems.
Spill switch witch is a heat sensor between your chimney and furnace.
Roll out switch
Inside the furnace neer the burner tubes.

If this system requires a fan belt then maybe it's just a fan belt.
Turn your system off and check the area of the blower motor. If u see a broken belt then that is the problem.

Your tstat may be bypass by another tstat. (tstat problem)
If this is a steam system the your pressure control needs adjustment
(on and off problem)