Ok, plastic off, quad standing up vertical I checked everywhere. I found this first, a black capped wire coming from the hi/low/reverse selector. I'm assuming it's for the reverse light. Using a tester it's hot. Following the wire, it runs about 4 inches then into a 2 prong plug, but there's no other wire going in or out.
http://i28.photobucket.com/albums/c212/Joshwfl/3.jpg
http://i28.photobucket.com/albums/c212/Joshwfl/4.jpg

Next I found this black capped bolt that looks like it should have a wire, but doesn't. It's right at the base of the cylinder on the left side, near the first picture.
http://i28.photobucket.com/albums/c212/Joshwfl/1-1.jpg
http://i28.photobucket.com/albums/c212/Joshwfl/2.jpg

The reverse light and neutral light socket both have 2 wires and both wires are hot and the lights don't light. When I jump the plug for the neutral light this relay clicks. What is it? It's located at the back of the quad near the selenoid.
http://i28.photobucket.com/albums/c212/Joshwfl/5.jpg

Also, I checked the starter button and both wires are hot all the time. Is this correct?

Now, like I said before I am not using and do not plan on using the electric starter. It doesn't work even with direct power and I'm not putting any money into this. I have spark when pulling it over, so I'm not sure if any of these wires, plugs or lights even matter in my situation. I need to go get the proper feeler gauges before I can check the valves, but when I pulled the 2 covers off today the action looked fine.

 

My thought on this whole deal.
Quit screwing around with the whole electrical deal. You said you have spark. This is how most neutral safety lock outs work. Some work on the starter system, some work on the ign system. If you have spark, you have spark; time to move on to something else.

#1 Do a compression test, like Hondabuster said. Get a guage and do the test. Don't cheap out and do a finger test; even 30psi will blow your finger off the hole. Post the numbers. Compression bad, find the leakage; Compression good, go to #2.

#2 Shoot a shot of ether down the carb. Kick it over, or better yet, try to push start it. Does it pop or start for a second or two and quit? If so, #3

#3 Pull the float bowl drain. Any gas in there? If no, follow it up until you have gas, then down to you find the blockage.

This is a simple flow chart and I don't have time to write down every scenario. You get the point and this will get you started.

 

Like others are suggesting it's probably not the neutral safety since it is sparking, but if the neutral safety works off the ignition like I think it does ( did on my 86 anyways) you will need to somehow bypass it to make it tow start. Jack up the back end and put it in gear and then try to pull the cord. Do you still get a spark? If not it won't start while in gear. I don't think any of your pictures are the neutral safety. The clicking in the back is probably a relay or a fuse of some type. Jumping the sides of the neutral should blow a fuse or kick a relay (not the same as crossing the neutral safety wires. Trace that wire back and see if you find the safety prongs. If it isn't something worse like the compression or timing, tow starting will be your best bet at making it run. The lights not working might just be bad bulbs. Good luck , sorry can't be of more help.

 

This is probably your million dollar problem right here..Carb not connected to the airbox?????!?!?! When that happens to my rap it is usually 1 carb knocked off a boot and since there are 2 carbs, one does different jobs in the rpm range. The right is for bottom end, and left is for mid and top. Single carb is for all of course. You need to get an airbox for it and a filter. If there is still compression it shouldnt be the timing chain, unless it doesnt have a timing chain. The carb is taking in way too much air. Try getting an airbox and filter...and if that doesnt work, then you can mess with the motor.

 

Did you ever get this machine going...

It sounds to me like a weak spark. You mentioned yourself that the spark was orange. Orange spark=weak spark. You want that baby to be blue. Have the coil tested.

 

I cleaned the coil ground and the spark advancer and got a bright blue spark. It still wouldn't start so I tested the compression and got 80lbs WOT. Started doing the tear down 2 nights ago and I have the valve cover off, the cdi is off, spark advancer out, but the spark advancer internals won't budge off the end of the cam. I can't get to the piston until it's out, but it won't move. I'm stuck at this point and am still working on it. There are 2 small clips holding the internals onto pins, I'm thinking that must be the ticket.