Quite a few things to address there. A simple one to answer is what the coil on the coin door is. That's a coin lockout relay. It's meant to keep you from putting money in the game when it's off, permanently tilted, or otherwise not ready for you to do so.

Let's work on your F4 fuse problem before anything else, though. J2 does provide ground returns for your flippers, so that'd be why they're not working. It's also appears to power the chimes, and also makes the test switch on the coin door work. And it effects the strobe on the MPU, which might explain the lack of displays.

Generally a fuse will blow because something is shorted. I'd check the coils on the chimes, first. They should read very low ohms between each lug, but not a dead short. If you can disconnect the chime box separately, I'd try unhooking it and see if the fuse blows.

If it's not the chimes, check your general illumination to make sure no light bulbs or light sockets are shorted. Also, that coin lockout relay and coin switches shouldn't be shorted either.

When J2 is plugged in you should be able to press the test switch in the coin door and go through diagnostics, seeing all the lights blink, all the coils fire, switches work, etc.

bally blackjack schematic.png
This schematic shows the F4 fuse, the rectifier, and the J2 connector on the rectifier/power board.

J2 to J2 on solenoid board and J3 on MPU.png
This shows where the J2 connector pins run off to on the solenoid and MPU boards.

No, they should all have diodes wired between them. They keep the collapse of the electromagnetic field when the coil turns off from going back up the line and damaging the MPU. You'd actually want to make sure they're all there and in working order.

I have a bit of phone anxiety, but I'll gladly help any way I can on here.

So when it started working for a couple minutes, did the fuse blow when something happened (pop bumper fired, flipper pressed, etc.) or was it just sitting idle?

Right, it should be a 5A. That could definitely explain it blowing after a couple minutes of play. Two amps versus five amps is a big difference. Just putting in the right fuse may fix that part of it.

The displays may not be working for a few different reasons. It could be a cabling issue somewhere back from the displays. We can try to trace that once the game is working a bit more stably. There is a high voltage section on the driver board that often needs to be rebuilt to get the displays functioning properly. There are a couple large caps there that dry out and need to be replaced. There's also a fuse there that could be blown, you might start with that. It's a really short, non-standard size fuse. You can move the fuse clip up a hole and install a regular size fuse for future ease of replacement.

Here's a picture of the high voltage section, with the fuse labeled: http://techniek.flipperwinkel.nl/ballyss/rep/blyhv1.jpg

Oh, just for the sake of speaking the same language, these games use charged gas displays, rather than LCDs. They have thin layers of Neon gas in them that get excited and glow when you run high voltage through them. They're fairly fragile and they wear out over time due to gas escaping from them.

That's great news! Especially glad to hear the displays are working now.

If you can get the test switch in the front to close, the game should go into test mode. Each time you press it it should go through a different part of the test. One part is a light test where it should blink all the computer controlled lights. When you're done with the tests just turn the game off and back on to get out of test mode.