The engine will never run without a battery. If there is no battery hooked up, the regulator will not put out any voltage. If there is no battery hooked up, you have nothing to power the anti backfire solenoid and the engine will run for a bit and die.
Look at the Regulator/Rectifier. There is one lead that is by itself. The other 2 are connected where they enter the body. Hook the stator up to the 1 lead by itself and one of the 2 connected leads. Hook the other connected lead up to wire going to the battery.
As for starting...make sure the choke is closing all the way. Possibly the gas is too stale to fire the machine up but not bad enough to keep it running once started?