Welcome to the board !! The problem with the horn may be the horn relay, contacts may need burnishing, its supply voltage may be low due to dirty wire terminals, or replacement is in order.
The internally regulated alternator is a good idea. Mad Enterprises has a good schematic for change over. I suggest you look for it here from previous posts using the Search function.
Now, for the horn, 5 VDC. Earlier suggestion, contacts corroded. Or a bad ground connection at the horn mounting screw, or a needed ground wire from the radiator support to the negative battery post.
Also, on the horn relay is a buss bar connecting the battery with the rest of the car. There is also a junction block next to the battery. Then lastly, the firewall plug. All these connections should be cleaned and coated with dielectric grease to prevent them from the elements.
You also might check the engine compartment for ground wires, braided wire actually, from the engine valve cover bolts to the firewall, and ground braids from the firewall to the subframe.
The dome light lens melting is normal, at least it WAS in my car, I removed the lens permanently.
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