I would think the cap is OK. Caps either short or open and you're indicating some capacitance. If you use a higher range on the VOM the test current will be lower and you'll see a slower response.
As for the microswitch and the parts that fell out. My guess is the switch is the starter switch and the thinggy that drives it broke.
The Klixon is an overtemp switch. Normally closed and opens at high temps. They usually have a temperature stamped on them but maybe not. Should read zero ohms when disconnected. If you don't disconnect it you may be reading resistance of a winding instead. Never had to buy one but Klixon has been around for years and should be a common item if you look in the right place like McMaster or some place like that. I doubt it's bad.
The centrifugal switch is closed until the motor is up to speed, then opens. It connects a separate winding with the capacitor in series. (usually) The starter circuit gets the rotor turning in the right direction. I'm thinking that if you measure the resistance of the microswitch it will be open. The centrifugal assembly usually makes physical contact with the switch until up to speed then pulls away so there is no friction and wear of the mechanical components when running. This could definitely be your problem.
On the two speed motors they switch another set of windings in to halve the rotor speed and I'm not sure how all this plays together. Never played with a multi-speed motor. In any case you have to figure out the jizsaw puzzle of broken parts.
Al