If you are referring the the area underneath the playfield, as I suspect, then the kicker is getting stuck on the surrounding frame. From an amateur perspective, this seems like the warning signs of the welds in that area weakening. With the solenoid getting stuck on, I can see the rod protruding enough to keep the ball from triggering the switches there.

Many possible solutions, but they all center around trying to get that kicker centered. Duct tape is the cheapest. Re-welding is the most durable.

I don't know that it's causing the rest of your issues, but it wouldn't hurt to try.

Not in the hole. My guess is that once it is fully extended, it is touching the frame of the container (likely the bottom). From experience, for whatever reason, the plunger gets magnetized to the frame and gets stuck there until the charge is removed.

If you run the solenoid test on the kickout with the playfield halfway up, then if this were the problem, you would be able to see it.