1998 Dodge Intrepid loss of coolent

The leak inside on the passenger side floor is from the heater box and yes the heater core is leaking. You never know when a thermostat could go bad but from what you have explained I would not be afraid to take it anywhere as long as I had some water with me. If the head gaskets are blown badly it blows smoke really alot. Fix the overflow leak though.
You could bypass the heater core. You say it leaks slowly when not running? Fix that leak if its not the overflow leak you spoke of.
After you turn the car off the engine cooling system needs a closed overflow system to replenish itself from any small leaks. And taking the thermostat out will definitely keep it from sticking also. But sooner or later the cooling system will have to be examined and pressure tested. The thermostat should be replaced and heater core would be nice for the winter months. The cooling ststem is so important to be clean and hold pressure because as pressure increases, so does the boiling point of the coolant. And thats why a small leak is not very good. Heat is the biggest threat to an engine.

Best to get a brass L or T fitting 5/8ths I believe, both my cars are bypassed. It is better, for a smaller problem then replacing but you miss the heat, in northeast. OK watch for your eyes, do it with safety glasses, cold. OK Joe