These things are the mutants of the moped world--- totally bizarre in the combinations of features and parts, hard to modify but so far very easy to work on.
I'm the kind of guy who has a natural feel for electronics-- and the QT 50 electrical scares the heck out of me-- it is a horrid little snake pit. I cannot understand how or why they would stuff so many relays and wires and lights under that control box.
"shorted the ignition circuit....?" uh oh. no good. How's the battery? not so bad on the QT but the jappeds can blow out headlamps etc with no battery--- it acts as an um... rectifier or something. chances are good that the battery is total junk, no sense in trying to salvage. Get a replacement.
These mopeds are 'power assist' starting. You have a much better chance of them starting with a good battery. You see how you have the "start" switch and then the "run" switch?
Mopeds tend to have rusty tanks and old ones, debris. Most people simply scrub the surface rust, oil the tank and keep a spare filter on hand. Running w/rust isn't terrible if you filter everything.
Don't use starting fluid, but put a bit of pre-mix into the plug hole. See if it fires. Make sure your ignition (is it cdi? hmmm) is fried for sure before doing other stuff.
This was just a quick FYI I banged out... a moped with a driveshaft?!? leave it to the Japanese.
Also, from what I've read other posters often mention the can tell their qt could "breathe" a little better. Make sure it's getting air and thhe exhaust isn't clogged. i thought mine was good until I put a mapp torch to it. yuck!
good luck, I'll keep adding what I can-- I love those QT's. There is a shop manual scanned as well-- that's awesome. i'm sure I could e-mail you the link. I'm the one who scanned the thing but I never know where it lives at any given time... LOL.