If the solenoid if firing, that means the switch at the dispenser is actuating when the paddle is pressed. I would look at the wiring around the motor and possibly behind the dispenser to make sure nothing is loose, shorted, or broken. The blue wire goes straight from the switch to the motor and then the other side of the motor picks up a common white neutral to complete the circuit. Start with the connectors and look for a pin not making contact and then check your wiring.

Those look like phantom voltages your meter is picking up. Try looking at the bottom hinge of the freezer door where the wiring harness comes through. Look for a broken or pinched wire. Basically you are loosing your source voltage somewhere and it's a matter of working back toward the source until you find where it is missing.

This was a problem with some of these doors due to the wire bundle being zip tied resulting in the wire bundle flexing when the door was opened. The best way would be to solder the wire back together and cover it with heat shrink or electrical tape. The only other option is to buy a new door as the wire harness is foamed into place during production. Once you make the repair and put it back together, remove any zip ties that restrict the movement of the wire bundle from the hinge or from behind the grill.

Glad I could help.