Steve,
More thoughts:
1. What is the
oil temprature doing?
2. In answer to number 5, yes but what a mess! I would check the water pressure. It will show you if the water pump is working and if the thermostat is opening. This is an easy and quick check. Much better than making a mess by removing the lower radiator hose or worse the water pump. This test would also show you if your radiator cap is working correctly.
3. In answer to number 6: Thank you. Did you check the fan? Even Toyota has recalls now and then.
4. By drilling a hole in the thermostat and bleeding the air from the water temp fitting you should not have much of an air lock. That is why you should always put the hole at the top. Do you have the small bypass hose that connects the water pump to the intake manifold hooked up or plugged? If you have the hose you should not have much of an air lock. The bypass hose is to keep the coolant moving in a loop while the thermostat is shut. The idea is to eliminate hot spots.
5. In answer to number 8: It is not likely that the water pump is bad. See above for easy check.
The thing that keeps bothering me is the spewing cold coolant. This is a problem. Check the radiator cap. I hope you don't have a blown head gasket.