I'd bet your transistor is bad. Test it with your DMM, or just replace it and the pre driver. It's rare for a coil to melt down that bad and not damage the board.

It could, but not common.

The slingshot switches aren't on the switch matrix, they directly activate the coil via the transistor. There are extra switches under the pf that the mech closes when it moves, and those are in the switch matrix and this show up in switch test. If you move the kicker arm by hand you should see it register in switch test, but that won't affect it firing.

I've occasionally seen where a transistor is 'bad', enough that it's just sorta 'floating', and then grounding something will make it float on instead of float off, or other weird stuff. Once something electronic has broken it no longer acts logically.

Almost definitely one of the ICs that's driving the 2n4401. If you had a logic probe and a test rom it'd be easy to track, but really I'd just swap the chip tjsg drives the 2n4401, and if that fails swap the next chip back, etc

For the schematic, look at the one for Phoenix. Same boards, but it's very good quality.

I would assume those would work for the chips, but I'm not an IC expert. So far, to me, what I've gathered is that the letters after the number don't really matter that much? If there's letters between the numbers though (eg 74LS02) that can be important. It's hard to tell on jameco since they don't actually give you a specific part