BMW Repair: HOT E46!! Thermostat or Water Pump??, transmission oil cooler, radiator hose

Question
Evan,

Kudos to you and your infinite knowledge!
Here is my problem: My '99 BMW 323i had the coolant expansion tank ruptured (according to my wife it actually exploded! Loud Bang! Coolant all over the road).
She turn off the engine immediatly and the engine runs normal as long as it is cold outside.
I have the expansion tank now replaced, top off with 50/50 coolant but the engine overheats within 5 minutes!
Engine Block and top radiator hose is hot, but radiator itself is cold.
How can I tell if my water pump is running (pumping) or if the thermostat is the culplit (by being locked and not letting the water out of the engine, or if I have an airlock?
Does it run without the thermostat?
Where is it and how do I replace it?
Does this car has two thermostats, one at the engine block and another for the transmission oil cooler at the bottom/base of the expansion tank? Should I have them both replaced?
Thank you so much!

Robinson
PS: No heat on the cabin vents also!

Answer
Rob, Thermostat is the black plastic housing with the electrical connection on top above the water pump that the upper and lower hose attach to. I would definately change this component reguardless of if it is the root cause of your problem. Look at the coolant in the expansion tank with the engine running. Make sure it has coolant and have someone rev the motor. The level should change as the pump changes speed. Fluid should swirl in the resevoir. Make sure the electric fan works. Also check the mechanical fan clutch for engaugement when motor is very hot. Bottom radiator hose should be hot and pressurized when the thermostat opens. The transcooler has a thermostat, but I have never seen one go bad or affect Engine temp. When bleeding the cooling system turn the A/C controls to hot and put on low fan speed. Make sure the Radiator cap holds pressure. The O-rings flatten out over time and no longer hold pressure. Good luck.
-Evan