Do you have the manual for the game? I can get the schematics from ipdb.org, but there's some info in the manual that might help us figure this out.
First off: Did these problems only begin after you fixed the drop targets? Do you have a DMM and soldering iron and know how to use them? (There's tons of help here if not)
I'm thinking that switch that came up in diagnostics might be stuck closed (aka making contact) and could possibly be the root of some of the other switch problems you mentioned. Does the manual tell you which switch the diagnostics were indicating? You said "Horseshoe" so I'm guessing it could be one of the 5 star-rollover switches that line the horseshoe loop. With the game off, I'd open the game up and get a close look at those switches--or whichever switch "090" is.
Make sure the wiring is properly secured to the switch terminals as well as the diodes. Wiggle the legs on the diode gently to ensure that it's firmly attached as well.
If you need to clean any of the switches, a business card or folded up piece of paper can be slid between the little contacts (the little round things at the ends of the switches that make contact with each other, carrying the electricity from one switch leaf to the other).
If a switch needs to be adjusted, the tool Colson posted is ideal. Definitely get one when you place an order for parts (I keep 2 because I constantly misplace them). In the meantime, you can use needle-nose pliers or a flat head screwdriver to 'pry' the leaves away from each other--but whatever you use to make adjustments to the switches--you want to bend the switch leaf at their *base*...the part of the switch leaf nearest the stack of brown spacers that the screws go through. Don't bend them at the end where the contact is!
Sounds like there's numerous issues, but I find it best to select one problem and work through it. Then see where you stand and proceed to the next problem. The 6 flashes on the MPU is a good sign, it tells us the game is booting properly.
If you need pics or further clarification for anything I've said, let me know. I remember when I first got started and didn't know squat about these games, so I will try my best to keep my explanations as simple as possible. I'm also jealous, btw...I've wanted a Sharpshooter for years... I will enjoy it vicariously once it's all fixed up! Good luck!
I've attached a diagram and added my laughably questionable ms paint skills to point out some of the jargon I used regarding switches.
vzv (resized).JPG
EDIT: I read your other thread about an issue with the start button and you said there was a connector that once wiggled got the game working. I'm betting you're gonna need to re-flow the solder to the header pins on the board that connector attaches to. Do you have any experience with circuit board soldering?