There is no problem adding coils with diodes as long as they are wired correctly (the fat wire normally goes on the stripped side).

So here is what happened, by reversing the wiring and applying voltage to the anode (non-striped side) you forward biased this diode, and as soon as the coil energized, the diode conducted as much current as you could give it right to ground, no current passed through the coil itself, so the coil is fine; however, this massive current draw immediately destroyed the diodes depletion region making it a dead short, the drive transistor P-N junction was also destroyed making it a dead short, and then the fuse blew.

You must replace all of three of these items before continuing.

EDIT...it sounds like your game didn't have a diode on the coil to start with, if this is the case, you can just remove the diode, rather than replacing it.

This is hard to answer, but I would tend to think so, assuming you can cut the transistor from the board, you can then simply heat the legs and pull them out one at a time; if you have a desoldering station, then yes, its very easy.

Based on the wire colors in you picture VIO/ORG and BRN/BLU you are talking about replacing the drive transistor for SOL14, This is Q48 on the PDB, which is a standard TIP 102.

Good luck!
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Yes, you should also have heard the coil fire as soon as the game was powered on.

You have to remove the board in order to properly solder in the new transistor.

There is nothing you did that could hurt this diode as its held reverse biased by the +50V source voltage. The worst thing you could have done is equalized the voltage across this diode.

Best thing to do is to just compare it with one of the adjacent transistors as they should all read about the same. Typical troubleshooting would be to measure from C-E as it should have high resistance when off, a failed transistor would measure a low resistance from collector to emitter (C-E) allowing it to conduct when off.

NOTE: The center pin on a TO-220 transistor is bonded to the metal tab.

Have you removed or replaced that diode on the replacement coil??

Please let me know where I failed in this post so I can be more clear on this in future posts.

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I've seen much worse, its an easy repair if you want it professionally repaired, or like zacaj mentioned the traces look good and while not ideal, you can probably get it to work if you can solder the top side legs to the traces; the downside here is that it any future repairs will be more difficult.

This is an example of a before and after of a botched repair that actually pulled pads AND traces off the board.

Before:
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After:
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Awesome!