Yes the pump is running all the time. You can run it with the thermostat out. The quad will usually run below normal operating temperature. The only side effect is it will burn more fuel and has the potential to foul spark plugs. You may also get a bog.
Ya, that fan should stay on all the time. Sounds like you might have a broken wire somewhere. You could unplug the fan and then unplug the sensor. Take a test like and make sure you have constant 12v at the fan plug.
Don't know what to tell ya. Seems like you have covered it all. Is the fan coming on when it gets hot? with the bike running unplug the sensor and the fan should come on. As long as the thermostat isn't stuck close then it should be good. I would pull the cover off the pump and just make sure the impeller is in good shape. Also make sure that the system is good and bled. Any air in the system can screw it up.