Thermocouples are a resistance type circuit. To test them you would disconnect both leads and measure resistance(OHMs) across the leads. If there is only one wire then measure to ground(outside of thermocouple body). The resistance should change in conjunction with a change in temperature.
No. Thermocuples are not resistance type, they are voltage GENERATORS.
The "type" of thermocouple is the combination of the two types of metal each leg of the thermocouple is made of when these two dissimilar metals are welded together (usually with a carbon rod), they create a thermocouple joint. This joint will create a voltage in relation to the temperature the joint is exposed to, which can be measured at the other end of the wire.
Since each joint of a dissimilar metal creates a new temerature measurement point, all of the extension wire and the socket where the thermocouple plugs ito the gauge or Analog to Digital Converter (ADC) must be made out of the same "type" material as the sense junction. for those who are familiar with electronics, you will already know that a circuit must be a complete loop, so there will always be at least two places where each type of dissimilar metal will have a junction - the sense junction and the measurement juntion, what is called the "cold junction". the tempeature of the cold junction is often easier and safer to measure with other means, so it is measured externally to compensate for the voltage generated at that junction (leaving only the measurement junction voltage, and thus the tempsrature at that junction).
Basic check you can do for your thermocouple is continuity. Check that there is continuity between both sensor terminals, and no continuity between either terminal and the engine. If the sensor circuit is open your thermocouple is bad.
If you want to rule out a gauge, k-type thermocouples are very comon, just pick one up and wire it in. Analog thermocouple gauges are self powered by the thermocouple voltage, digital (ADC type) are powered by the vehicle ignition circuit.