Yes; I also found out last night that the coil in the one they installed is meant to be horizontal and not the v coil he spoke about. 

Many people automatically think electric resistance baseboards (or electrical coils in a central air-handler), but your ancillary heat could be easily fuelled with natural gas, propane, oil or another fuel.Presuming your use of "furnace" means a forced air furnace and not a boiler and hydronic heat, you have the option of adding a heat pump with an exchanger insert in the plenum of your existing ductwork.  

Someone formed those strips off a roll of vinyl-covered aluminum trim coil and caulked them into place when the old window was replaced.

Here in Michigan it might be different, we have a coil installed in the furnace which allows it to be separate thermostats.

A picture of a hole in an a/c coil the size of a screwdriver goes a long way.

*Check the functioning of the refrigerator and vacuum the coils and clean the drip pan; also wipe clean the back of the refrigerator. 

Get the radiator/oil type, not the fire-hazard exposed-coil type.

My estimate: 2-3k New HVAC Unit (not just the unit outside), Coil, ... etc will need work.

If it’s the former, replacing the ac and coil should be in the $5k range.

I would pull the cover off the a-coil and see what's going on in there.