I have a 1983 QT and it will run without the battery. The battery is only to help keep the lights going. AS far as anything else i haven't torn mine apart that far so i'm not too sure. I know there is a wire to the coil. Make sure thats connected. Also make sure that your kill switch on the bar is in the on position. If you can't visially see anything you can always test your parts to see if there bad. Look for any loose wires that might be touching the frame and maybe shorting the ignition out.
If you go to www.mopedriders.org and go to maintenance you can find the shop manual in there in .pdf format. Its shows you how to test all the ignition parts. Hope this helps.
One more note. Are you trying to start it in the run postion on they key? I have a hard time right now with mine. If i start it with the key in the start postion if idles fine. I jsut can't rev it up. Which is normal. Once its warmed up if i switch it to run it will kill the motor. Seems like its shutting off the spark. if i flip the key a few times from run to start it will eventually stay running. I took the ignition switch apart and found no bad contacts so now i'm in the process of checking the wires. I'm pulling the whole bike apart anyway. If thats what your problem is then your in the same boat as me. not sure why it does this. If not just double check those ignition parts with a meter like the manual shows you. Theres not too much for the ignition. From what i've read it not points. So you should jsut have a pick up and coil. most of the other electrical runs the lights and stuff.