Heater blowing cold air - coolant leaks when turn engine off
 

I have a 2002 Chrysler 300M. After turning off my engine the coolant fluid started to leak under the middle front of the car. The temperature gauge did not ever get hot. After car cooled, it stopped leaking. The next time I took it out I was idling for about 10 minutes (after I'd been driving about 20 minutes) and realized the heater was blowing cold air. Then when I took it back out on the road the temperature gauge started to get hot. I pulled over at a garage before it overheated. Small puddle of coolant under car after I talked to the mechanic. When turned the car on, no leak, still blowing cold air, temperature gauge normal. Ran for about 15 minutes no leak and temperature normal, turned off car and started leaking about 5 minutes later. I've heard a number of theories but none that can account for all the symptoms. Any thoughts appreciated.

u have a coolant leak plane and simple.
it will only leak out when the engine gets hot and the fluid is under pressure.
Have some place do a pressure test to find the leak.

Pressure test done-still have questions
 

Had the pressure test done and they told me they 'think' it's the water pump but won't be sure until they take it off and at that point I should just replace it because the labor to check is the same as the labor to replace. I'm concerned if that's not it, I'm out big bucks and I still have a problem. They said the leak was coming out by the timing belt cover. I've been told a few different things.
1) If they did a pressure test they should know exactly where the problem is, not maybe
2) If the heat was blowing cold air it could be the thermostat was stuck open, would the pressure test, test for the thermostat?
3) The heat may blow cold air if there's air in the system and the leak could be coming out the weephole because of the pressure build up.
Do any/all of these scenarios make sense?

Thanks for reading.

both of these are irrelevant due to a leak low coolant will cause no heat until driving.
Weephole if the pump was bad like a wobble if the deal is leaking comes out at lowest point.
get the pump replaced.
If the heat was blowing cold air it could be the thermostat was stuck open, would the pressure test, test for the thermostat?
3) The heat may blow cold air if there's air in the system and the leak could be coming out the weephole because of the pressure build u