If the bike is in neutral it should start with the kick stand down and the clutch lever not pulled in. Check the handle bar switch with the start button and kill switch, there could be some corrosion in there that would cause problems and it is easy to clean. Headlight not working and the blue light not showing could be a plug connector behind the headlight in the rubber boots. See if your headlight is in fact not burned out. You should be able to hook it direct to a 12 volt battery to check it. I can't help you with the purple wire and have no clue about that.
I have a sticky posted above regarding Bringing back a sitting bike with some good advice for all of us. I have brought back some real basket cases by following some simple but time consuming procedures I mapped out in that thread. There is only one fuse that I know of behind the right side cover where all the wires are located. If any of the electrics are working then your fuse is good. If your fuse is fried then nothing will work. Search through the posted stickys above and you can find the wiring diagrams and it could help you trace these issues. Some tanks can be cleaned of the rust inside while others could be too far gone. Lots of threads here on tank cleaning and internal sealing but most important after you do get it cleaned up is installing a small fuel filter in the line from the tank to the carb. Clean the carb as I am sure it has gunk and clogs and very important to check the seal around the rubber boot between the carb and air box. These boots shrink and can cause all sorts of running issues if not tightly sealed. Mice love building nests up in the air box right behind the battery where it draws air from. We have dissected these old style carbs in every way known to man and the slightest blockage will give you fits. Tons of info here on this finicky issue for sure and some get very lucky with just a good cleaning while others go through the pains of hell trying to get the bike running right. I just gave up on a couple old carbs and bought brand new ones and solved the problem immediately.
With the electrics I go through a process with my multi meter checking each wire from one end to the other to make sure there is continuity and no hidden breaks. Those bullet type push in connectors can get corroded or have a loose fit and all the multi connectors could need some cleaning, all it takes is just one bad connection or a bad ground anywhere to have you scratching your head. Been there and done it and finally bought a complete wiring harness and replaced the entire mess to get it working. I hate electrics! The 1988 through 2000 TWs are some of the very best ones IMO and once you get it all figured out they are great bikes. There is a couple wires that come from the left side engine cover, Stator, and go up to the battery compartment. Some guys tend to pinch these wires when they change sprockets and put that engine cover back on by not routing them in the tiny slot correctly. It is probably safe to say if you have spark at the plug then this is not an issue on your bike. Just checking for spark at the plug is a good indicator that the bike should run. Key on and the kill switch in the right position and you should see spark. One step at a time doing all the checks should get you up and running and just about all the info you could ever need can be found right here in the technical section, stickies and the posted write ups many of us have gone to great lengths in posting.