Probably not unless so much coolant has leaked into your passenger compartment the engine doesn't have enough left to keep it cool. If that where the case you would definitely know. First thing to check is the simplest thing and that is your thermostat.
No
yes.
you would have to bypass the heater core first.
low coolant? Bad thermostat? Climate control not sensing proper temperature? Heater core plugged or restricted? Heater core airbound?
There are several things that could be going on. The most common thing that would make your car smell like gas when the heater is on would be a bad heater core.
No it wont. The heater core is just a small radiator inside the cabin that when the fan is turned on heats the air. A bad heater core will cause you to have no heat inside or will leak coolant on the carpet.
You probibly have a bad heater core.
Bad or clogged heater core. Backflush it.
try changing out the heater core or the therma stat may be bad
Bad window or door seal? Bad heater core? Plugged evaporator drain? Came home drunk last night and parked it in the living room aquarium? heater core went out
the heater core, heater core hoses or heater control valve is leaking
coolant leak inside car underdash on pass. side