MY Chrysler 300 M IS OVER HEATED AND IS RUNNING HOT

I'd start with a test for a leaking head gasket. If combustion gases are sneaking into the cooling system, it can pool under the thermostat and prevent it from opening. Thermostats open in response to hot liquid, not hot air.

Another clue is you might see air bubbles in the reservoir when the engine is running. The leak test involves drawing air from the radiator through a glass cylinder with two chambers partially filled with a special dark blue liquid. If combustion gases are present, the liquid will turn bright yellow.

Auto parts stores sell them. There cheap and work well. Thay might call it a block test kit

If the thermostat is closed, no coolant circulates so no hot liquid enters the heater core.

You can borrow the head gasket tester from an auto parts store that borrows or rent tools but they will make you buy the special liquid. The liquid will become contaminated if you accidentally suck up coolant during the test, then it must be replaced. To insure you don't get a tester contaminated by the last person to use it, they make everyone buy their own liquid. Since you're buying way more than you need, it's less expensive to just have a mechanic who already owns this tool do the test. It only takes a few minutes.

Yes you can but you have to remember that some leaks only show up when the engine is hot and cracks expand, and some only show up when the engine is cold. There is a tester called a "cylinder leakage tester" or "cylinder leak down tester" that does just what you're asking. You must put each cylinder, one at a time, at top dead center on the compression stroke, then compressed air is supplied to the tool, and air pressure regulated to around 30 psi is forced into the spark plug hole.

You look for the results of any leakage in four places. If you hear hissing at the throttle body or carburetor, you have a leaking intake valve. At the tail pipe; a leaking exhaust valve. Leakage past the rings will show up at the oil cap or dipstick tube. A cracked head or leaking head gasket will cause bubbles to show up in the radiator. With relatively small leaks, the air will sail right past the thermostat and be forced into the reservoir by normal cooling system pressure and you might not even know there's a problem. It's the larger leaks that can cause the thermostat to close.

Many thermostats have a small bleed hole to help hot coolant get from where it became hot to the thermostat where it will be sensed and cause the thermostat to open. During the leakage test, air from the combustion chamber sneaks through that bleed hole and goes to the radiator. For thermostats without that bleed hole, air must find another way, usually through the heater core, so it can take a while to show up where you can see the bubbles. You can watch to see if the coolant level is rising in the radiator, but the tester also has a gauge to show the percentage of leakage. Typically anything under 10 percent is acceptable. If you find much more than that in one cylinder but no other hissing is noticed, give that one plenty of time for the bubbles to show up. The air has to be going somewhere. Also listen at the spark plug holes for adjacent cylinders. A head gasket can leak between cylinders, but that isn't what your symptoms suggest.

Removing the thermostat is counter-productive. The coolant moves way too fast and doesn't stay in the engine long enough to get hot, therefore you'll get cold air from the heater. Running too cold also promotes sludge formation in the oil and accelerated engine wear from parts that haven't fully expanded to fit properly, mainly pistons.

In some instances the opposite can happen. The hot coolant won't remain in the radiator long enough to give up its heat so hotter than normal coolant goes back into the engine.

Some water dripping from the tail pipe is normal. That is one byproduct of a working catalytic converter. If you suspect it might be coolant you're seeing, you can add a bottle of dye in the radiator, then search a day or two later with a black light. The dye will show up as a bright yellow stain. If you see that at the inside of the tail pipe, coolant is indeed going there.

I never tried that so I don't know what results to feel or expect.

Do the dlock test with the fluid that changes color. Its sounds like a bad head gasket or head.

Make shore you purge all the air from the cooling system. Try rasing the front of the car on a jack

Make shore you keep the surge tank filled with coolant so as the engin is cooling it draws is coolant and not air

Hard to tell by the pic. Make shore the bottle is kept full. It may still purge some air. Hope it works out