Engine getting warm

[UberRoo]

It's probable that you'll need more than water to open those thermostats. The water may boil before they open. Use cooking oil - it has a sufficiently high boiling point, plus it's cheap, easy to clean up, and clear so that you can see the thermostat open.

[CDG]

Cooking oil is good, but if the stat is sticking, keep your face clear, it can pop, spraying hot oil out of the pot. OK is some situations, not so good in this one.
Water should be ok, depending on the rating, a 195degree stat will start to open slowly aroung 190, and should be about half way open at 193, and fully open at 195. as the water cools down it should close at the same rate. If it moves sudenly, or sticks, or it looks like its moving on an angle, then ist bad, obviously if it doesn't open, its bad, but your issue is intermitent, so look for the earlier signs.
I'm pretty confident its the stat, but its an easy test to do.

[UberRoo]

Right CDG; cooking oil can ruin your day. All the bad thermostats I've ever tested always seemed to open well over 212 degrees which always makes me wonder if I'm doing something wrong - until it finally opens. Kinda reassuring to know just how bad the themostat is.

Also, regarding the reason removing the thermostat can damage an engine: The reason this can be harmful is because on some (or many) cars, in the thermostat housing is also the plumbing for the heater core. With the thermostat removed, the flow path changes and the water pump will circulate the coolant every which-way. As I recall, the pump typically circulates the coolant through the heater core and radiator - the coolant in the engine stagnates and...well, you can guess what happens. Having the coolant flow too fast is almost never a problem; however, it's academically notable that the coolant passages are designed for a certain flow rate. The gross thermal conductivity of parts in the engine toward the latter half of the coolant passages is higher than the gross thermal conductivity of the first half of the coolant passages. Because the coolant is heated along the way, the thermal conductivity of the components is increased to compensate for lower temperature gradient, which reduces the heat flow rate. If the coolant (even very hot coolant) flows too fast, it will under-cool the first coolant passages it passes through, but it will over-cool the last half of the system. This is still all academic because most cooling systems are minimum competency designs, which run at near-peak capacity all the time, and the engines can withstand large heat differentials without consequence anyway. Regardless, if your engine produces more power (and heat) than factory, a higher rate of coolant flow would probably help considerably with cooling.

[CDG]

Ya, what UberRoo said