What does the drop target bank look like under the playfield? Anything broken? Do the rest of the targets come back up?

They should all reset in unison. Does the center target not come up at all when they reset, or does it just drop down immediately?

Most likely something is broken then. There should be a ledge that all three targets reset on. The coil pulls the ledge upwards to push all three targets back up, so if the target isn't going up then it must not be making contact with the ledge anymore.

There is a single solenoid in each drop target mechanism that resets all 3 targets into the "up" position. This solenoid has a triangle shaped piece of metal with 3 tabs, one for each target. It is possible one of these tabs broke off. Lift the playfield and have a look!

Old Sytle: http://www.actionpinball.com/pics/01-7036a.jpg

New style (performs the same function): http://www.marcospecialties.com/pinball-parts/01-7036

I recently had this same problem with the lower left bank on my BK, the middle target was jammed and not moving up or down at all. Not sure if this was the entire cause for it, but the target's spring had somehow popped off the metal tab on the target housing. Gave me an excuse to remove, clean up and tune up the entire bank, and now it's back to working like a champ. In fact, it's working so smooth now that the other three banks seem a little balky in comparison.

Here's a diagram: http://www.planetarypinball.com/reference/partsmanuals/WMS_Parts_1980/files/mobile/index.html#117

In those years the mechanical stuff was in a separate parts manual.

If your flippers are weak can't hurt to rebuild them with new parts. There are kits available online.

Most of the lights on the backglass are General Illumination lights that should come on as soon as you turn the game on (like the ones under the plastics on the playfield), and aren't controlled by the CPU. There are some other lights (like Game Over) that are CPU controlled. Which lights exactly are/aren't working?

Not sure where to get the original ramps but someone on here is selling this mod: https://pinside.com/pinball/market/shops/1080-the-mod-couple/00866-new-custom-black-knight-illuminated-mod-replacement-ramps

You could also remove the ramp (4 tiny phillips head wood screws - 2 at the top and 2 at the bottom) and bend it back into shape, and screw it back down.

On the light issue, have a look at the manual, second to the last page should be the Lamp Matrix and the Switch Matrix tables. The Lamp Matrix table show the Same Player Shoots Again and High Score to Date are in the first column. Same Player Shoots Again is in Row 1 and High Score to Date is in Row 6. The other lamps that are not working are on Rows 2, 3, 4, 5, 7, and 8 so those signals are not making it to the backbox. Since other lamps on the playfield that are on these rows are working, we know it isn't a driver board problem. This leaves a wiring problem as the likely cause. There is a small connector in the wiring harness from the backbox to the light board. With the machine off, disconnect this connector and have a look at the pins to make sure they are not corroded and are physically strong. Plug it back together and make sure it is fully seated.

The whimpy flipper could simply be a poorly adjusted End of Stroke (EOS) switch. This switch engages the "hold" coil so the flipper coil doesn't burn up if you are holding the flipper up for long periods. To adjust it propoerly, push the pawl into the coil with your finger so to where it bottoms out. Do not push the pawl in by holding the flipper bat up or you will not end up with the best adjustment as the the slop in the links and such will cause you to end up too loose of an adjustment. With the pawl bottomed out, adjust the EOS to be a gap of 1/16 to 1/8 of an inch. If this gap is larger than this, you will notice reduced flipper power.

If the light blinks on and off then it's not a GI light, all the GI are wired together so they can't be blinked by themselves. The GI is controlled by a relay in the back which is fed by the yellow wires coming from the power supply board. Often these wires or their connectors are melted which could be causing your problem.

The GI wiring is fed to the backboard through the same connector as the controlled (lamp matrix) lamps. Sounds like this connector may be loose.

That switch could be a bit closer but it's alright. Looks fairly pitted though, so it could probably use a replacement. Flipper rebuild kits will include it too.

That black connector is the right one that connects the controlled lights and the GI to the backboard.

It is hard to tell from that pic but are those pins dark and corroded? They shouldn't be green at all and should be clean metal colored.

Also, there are 2 GI fuses, the one on the power board (which is good or all GI lights would be out) and the one on the back of the backboard (you can see it in your picture).

That is the GI power input from the transformer into the power board. It is regularly burnt. One of my boards got so hot there, it unsoldered this connector from the board!

Since it is working (you have GI lights on the Playfield), I wouldn't worry about it for now. If you are planning on swapping the GI lighting to LEDS, you can leave it as the load on that connector with LEDs will be significantly lower. If you are staying classic, you should replace both the part on the board and the pins in connector on the wire side. No need to replace the board, you can replace the connector that is on the board.

Did you check it with a meter, swap in a known good one, or eyeball it? Sometimes they burn out on the end and look ok when they're actually bad.

I would check those trough switches. If one of them isn't opening and the game thinks there are still two balls in the trough it might keep trying to kick one out. Could also be ball 3 is getting stuck and keeping that switch closed until one drains and pushes it down. Switches 41, 42, and 43 maybe? Look at switch test before a game is started. Which switches are closed? Start a game, grab the ball and hold it in your hand. What switches are closed now? Now go into two ball multi-ball state. What switches are closed now?
Pinwiki could help too: http://www.pinwiki.com/wiki/index.php?title=Williams_System_3_-_7#General_Illumination_Connection_Design_Flaws