If your HVAC mix is set to cold, then there isn't going to be any coolant flowing thru the heater core. If it's note set to full-cold, and there's coolant flowing to your heater core, then the excess prssure theoretically might cause the leak you're seeing in the cabin.
Changing a headgasket is relatively simple, albeit time-comsuming. The total cost, if you do it yourself, would be the cost of the new gasket, the cost of new headbolts(unless you're running studs), and perhaps the cost of some copper spray. If you work quickly, you could have it done in a few hours, but if you've never done it before, it'll probably take longer.
When you shut the engine off, the hot coolant remains pressurized for a while, until it cools down. Once it cools, it contracts. Depending on whether or not the contracting of the coolant pulls coolant from the reservoir back into the radiator, what you're seeing after letting the car sit may be vacuum in the the system, and not pressure.
It is very possible for you not to see any smoke at all, and still have a blown headgasket. On the intake stroke, the intake valves are open, so the vacuum may not necessarily pull coolant into the combustion chamber as you theorized, since the open valves are a path of less resistance.
If I were you, I would do both a compression test and a leakdown test to see if they can shed any light on the situation.