Excess heating

If you mean the coolant was being pumped out of a cold engine just from starting, that is not a good sign. It shouldn't do that until the coolant gets hot and starts to expand. If your kit is a pressure test kit, install it on the radiator on a cold engine, then start it and see what the pressure does. If it starts to rise instantly you have either a defective gasket or a crack somewhere, either block or head.

Pressure starting immediately isn't from a blockage. It is from combustion gasses in the system. Before you swap the radiator there is a simple test that can be done to see if you have an internal problem. It is sort of a reversed compression test. You remove the spark plugs and apply air pressure into each cylinder with the piston in a spot where both valves are closed. Usually this is done at Top Dead Center. It is like a leak down test but doesn't require a tester, although using one wouldn't hurt. You lock the engine from turning and apply 100-120 psi air into the cylinder. Then look into the radiator for any bubbles. If any show you have an internal engine problem. I like to also check with the piston lower in the cylinder but that can be difficult without taking the camshaft out to be sure the valves stay closed. What this does is put the same pressures that compression has inside the cylinder and you can find out if there is a crack anywhere. Even in the block itself.

Nope, if it still had air in it it would actually react the opposite way as the air takes longer to expand from the heat. That is why an air pocket in the system causes problems, it doesn't allow the coolant to get to the parts and can cause steam pockets if the engine does get hot and the coolant suddenly hits the hot surfaces. However that takes a while to happen. From a cold start it should take 2-3 minutes for the coolant to even start to get real warm. The only pressure in the cooling system should be caused by the expanding coolant. Nothing else causes pressure, the pump just circulates the water. From a cold engine start the coolant starts to expand as it gets hot, in a modern system the pressure increases until it either overcomes the pressure cap on the radiator and gets pushed into the reservoir or it expands in the coolant bottle. If you take the cap off and start the engine the coolant should set there until the engine warms up. Any pressure in the first minute is not a good thing. I just don't want you to replace the radiator and then discover it still has a problem.
You mentioned it had this issue before you replaced the other parts. If it got very hot there is a possibility of a crack in the block or the head. You say you rebuilt the head, did that include checking it for warpage or cracks?

It would allow flow but shouldn't cause it to overflow. But put the thermostat back in and see what it does.

That would be about the time it would take for the coolant to start to get warm, in a full system that is working okay it sounds about right for it to fill the neck and start to leak as it expands. But it shouldn't come pouring out.

Let us know what it does.

Modern engine run about 210 degrees or even a bit hotter. I wouldn't worry unless it truly overheats. Then I would check the cap as well because they hold pressure that raises the boiling point of the coolant.

It could be the gauge, the only way to know that would be to measure the temperature with a mechanical gauge in the same spot and see what it shows. However if it got hot enough to lock up you may have other damage internally. While it was that hot was any water or coolant added to it?