It will run with the coil test in diagnostics. Also check the fuses on the shaker motor board right next to it. This is where to start IMO. After that you will have a more finite question.

You can also test the windings of the motor itself using a resistance test from the + to - lugs of the motor itself. This would tell you if the motor coil itself is open or shorted. You should see a very low resistance reading like 2-6 ohms but not 0. A 0 would likely blow the fuse.

Yes, but you would have to somehow attach or hold your multimeter probes on the +and- motor connections with the multimeter in AC voltage test. Then go to the shaker motor test (does this even exist?) to test for voltage. Obviously there isn't going to be any voltage when the shaker motor is not supposed to be on.

I believe Stern shaker motors run on 24V AC.