We currently pay for the gas that runs the current gas water heater and boiler that provides the radiant heat in the winter.

Heating mat underneath bathroom tile not working, need to replace and relay tile. 

Thats about the same as a cement mixer driver , or a heating oil truck driver 

I had a renter client who had a $5,000+ loss covered (frozen heating pipes & repairs made apartment unable to be used for an extended period of time.  

You could end up with a bigger problem if she ends up sick from the heat.

The item you are referring to inside the closet within the house is the air handler and is usually a large metal box containing a blower, heating or cooling elements, Let me add that if the air handler looks old, you may want to have someone do a system check on the whole system. it's usually inexpensive and could save you if all that's needed in cleaning up.

I have saved up and inherited enough to make a down-payment on a house here, but things feel over-heated ($1,000+ /sqft!).

Hot water heater, stove, heating HVAC and kitchen sink were taken.QuestionsShould I file a claim with the insurance ? 

One unit has electrical heating, the other has gas.

There are only a handful of us nuts willing to deal with the cold - both in terms of looking at houses (the heat is never on) and getting rehabs going (takes some time to get the heat on). :-)