Re: Thermostat
IMO if you can touch it while it's running you're just fine.
Why not pop off the thermostat covers one at a time and check if water is flowing through them?
Here's what I did recently for mine with no thermostats thanks to previous owner- From cold start on the muffs, pull off one thermostat cover (noting of course whether the previous owner removed it and then doing something about it) and start it up. If no water put it back on and pull the other one and start it up and see what happens. At some point water ought to come pouring out of an open cover. Note that my do one at a time advice is purely from my one cause/effect experience. As long as you have water on the impeller don't worry about letting it go for a bit to be sure that it managed to move the water up the tube it could take a bit.
If you can touch the head while it's running hard on the water and it's peeing and you know from removing thermostat covers that it's actually flowing water through them (peeing doesn't necessarily mean that water is going through the heads it just helps you tell whether your water pump is out of action) then I wouldn't worry about it.
Also if you don't know the last time the impeller was replaced, replace it.
