Engine cooling

You don't fix anything by removing the thermostat. In some cases that can cause overheating because the hot coolant doesn't stay in the radiator long enough to give up its heat.

I don't know which engine you have, but if it uses an electric radiator fan, be sure that is turning on and the coolant level isn't low. The next step is to have your mechanic perform a chemical test at the radiator to check for a leaking cylinder head gasket. That 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.

Rule the head gasket out first. That test just takes a few minutes.

Normally the thermostat keeps the coolant in the engine long enough to get it up to the right temperature, about 195 degrees. That's where parts expand to fit perfectly and the oil can carry away contaminants to the filter. While it's not common, a missing thermostat can cause overheating, especially with really big engines.

Feel across the radiator to see if all areas are warm. If there's a cold spot, some tubes could be plugged. More commonly, if the coolant got diluted with too much water over the warm summer months, the radiator could be frozen.

I really hate winter / cold / snow!

Oil and coolant mixing is the least common symptom of a leaking head gasket. There's 15 pounds of pressure in the coolant passages and about 40 pounds of oil pressure in the oil pressure passages. 0 pounds in the oil drain-back passages. There's WAY more pressure in the combustion chamber, and when you add the corrosive nature of ethanol, a leak between the combustion chamber and coolant passages is by far the most common place to leak. Also, each cylinder is surrounded by coolant and its passages so there's a lot of places the leakage can occur. There's usually only one or two pressurized oil passages and they aren't close to the hot cylinders.

Water pumps that fail to pump coolant are very rare. Only VW has had a lot of trouble with that because they used plastic impellers that crack and spin freely. If the water pump isn't circulating the coolant, it's more likely its drive belt is loose. I've seen a loose timing belt allow it to slip over a timing belt-driven water pump pulley. That can happen with serpentine belts too if the spring-loaded tensioner pulley is stuck in the retracted position, but those will usually squeal too.

You can drive it as long as it doesn't overheat. No hot air means no coolant is circulating through the heater core. Check how far the upper radiator hose is hot and whether the top of the radiator is hot. If they're all cold, the coolant isn't moving at all.