Based on what you have said, there is no way that anyone can tell you how to fix this without actually seeing the car. :nogrinner Your wiring is, no doubt, all hacked up from the installation of the alarm system, and, if big metal stuff has actually started melting, you also have some fried wiring in the harnesses. If you intend to keep this car, you will end up rewiring the whole car at some point in the near future, so, this is as good a time as any to start. That, however, is not really a cheap project, so, I can understand why someone would prefer doing this one stage at a time.
To get the car to where you can start it and drive it around with no anything electrical, you will need a new starter solenoid. You should have a fat pink wire plugged into a red/green wire coming off the back of the ignition switch. That pink wire should also unplug near the firewall. Unplug it near the firewall and run a wire from the end of the pink one out to the positive side of the coil. There should be a red/blue wire coming out of the back of the ignition switch. Run a new wire from there to the 'S' post on the front of the starter solenoid. From the 'I' post on the front of the starter solenoid run a wire and attach that also to the positive post on the coil. Disconnect everything from the threaded stud on the back of the ignition switch, unplug the headlight switch, and pull all of the fuses from the fuse box. What you should have now is a car that will start and run, but nothing works. No lights, no horns, no nothing. But, it also shouldn't be trying to set itself on fire.
When you get the car to this condition, turn the key off, disconnect the negative battery cable, take a volt meter and put the positive probe on the negative battery cable terminal and the negative probe on the negative battery post. There should not be any voltage showing. If there is, you have a short along the main power feed that goes from the hot side of the solenoid to back behind the dash, splits off into three wires and takes power to the ignition switch, the headlight switch, and the emergency flasher circuit.
If this does not show power, then you can slowly start adding accessories. First, with the negative battery cable disconnected, plug in the headlight switch, and repeat the test to see if you have voltage between the negative battery post and cable terminal. If no, good. If yes, make sure that you don't have the parking lights on, or the glove box light on, a door open, etc.. Go through the car, system at a time and bring everything back on line, repairing stuff as needed.
Definitely remove all of that alarm junk and repair whatever damage was done on the installation.