you most defiantly have a bad coil, and bad Q transistor, got stuck on melted the coil, that why it won't pull in.

Prob not, just the coil and sleeve should do that. Transistor on MPU it tip102 and may have also taken out the predriver and IC but you can test them to be sure

You may need to rebuild the power supply, but first check the slam switch under the play field and the one on the coin door. If you are nudging the pins and they are set too close the game will reset to attract mode.

Check out Vids guide on rebuilding pop bumpers, its a must read for anyone.

Definitely buy everything new. Frankly, I would fully rebuild all three. If one has worn enough to freeze, it's time to do them all. I myself once took a shortcut on my pops by only replacing visually worn pieces. It didn't last. All those parts accumulate teeny bits of wear which add up to enough unwanted movement to cause faulty operation and worse. In my case, the price I paid for my partial pop bumper rebuild was a stuck pop which fried a transistor on the mpu. Then I got to do the pop bumper job again (correctly, replacing everything) plus repair the mpu.

Definitely read Vid's guide to rebuilding pop bumpers and do a thorough job on all three.

Congrats on the purchase. I'm restoring one myself and I'm done with the boards and playfield components. Everything works fine now so it's time to rip it all of and get things cleaned up.

I'll follow this thread and see how things work out for you. I think the inputs already given has set you off in the right direction. So I'm waiting to hear if your problems gets solved or new inputs are needed.

I've just taken off the bumpers and I see that they will not install easily when assembly time comes up. But I read through Vid's guide and one of the replies points to this alternative solution. I don't know how well it works, but I'm going to keep that in mind.
Link: http://www.edcheung.com/album/album08/pinball/tz.htm

Good luck!