Brand New Weil Mclain Gas-Fired Water Boiler not heating zones
Probably still have air in the system.

The circulating pumps on these do not provide much head pressure and cannot overcome significant accumulations of air in any vertical return lines.

If you have baseboard radiators with air bleed valves near the high spots then crack those open to vent until no air and only water comes out (with all zones circulating and also assuming your make-up water connection works correctly).  If you do not have the luxury of strategically placed accessible air bleeds then someone will need to properly flush air out by closing the loop across the boiler and flushing line pressure water in one side at high velocity while draining it out the other side across all zones until all the air is purged.

The separate heat/cool thermostats are likely because the original installer did not want to deal with integrating separate boiler and a/c controls into a single thermostat, which would have required slightly more complicated control work than is typical.