Sounds like your moped is not wired properly. We are shooting in the dark but try this. Take it a step at a time.
Look at the back of the tail light where the wires go into the unit. You may have to remove the tail light assembly enough to get to the back of the tail light.
There are probably three wires. If there are three one of them should from a tail light terminal stub to ground. Make sure the ground wire is grounded on a bolt, fender or whatever it is grounded to. Make sure this connection is not corroded or rusted.
Now only deal with the other two wires. Swap them on the tail light terminal connectors.
leave teh tail light hanging or tape it in place with some good ole duct tape.
Now make sure everything is set like you were going to try to start the bike.
Pedal it and see if the proper tail light comes on or if the brake light comes on when you apply a brake lever.
The popping the head light problem
There are at least two coils inside the magneto, one for the high tension coil/spark plug the other coil is for the headlight/tail light/brake light/horn. There are at least two wires coming out of the engine originating from each of these coils.
Each wire from the engine connects into a wire harness, one wire harness going to the light circuits and the other wire harness to the ignition circuit (Hi Tension coil/spark plug). Sounds like these two engine wires may be plugged into the wrong wiring harness. This would cause too much voltage to go the head light circuit when you pedal it and thus blow out your headlight.
Try switching the two wires that come out of the engine with the two different wire harnesses and then try to start it with a known good headlight.
You really need to get a manual on your model Peugeot with a wiring diagram.
Some Peugeots have 5 coils in the magneto further complicating the wiring complexity, thus you need a wiring diagram.