Assuming you have a new or properly repaired wiring harness because you said it was fully restored. You may be in an OK position.

If the harness is old and it's just a cosmetic restoration it could be a crap shoot.

I can sit here and tell you from your description It seems like several things are happening at once namely a bad horn switch. Also an old battery & voltage regulator caused the death of the fusible link/other wire. Those things can easily cause all your symptoms BUT it's best to start looking at the whole of it because electricity has a mind of it's own.
The fusible link is the connection of your cars electrical system to the battery. If it's bad it can cause all kinds of power fluctuations that can affect gauges, lighting malfunctions, random stalling when electrical devices are turned on etc.

Start with #1 That battery is 5 years old. I suggest for the diagnostic portion at least replace it with a new or known good battery.
#2 Inspect the fusible link. Sometimes when the fusible link starts to go it can cause intermittent connection between the battery and the alternator/under dash wiring. The Ammeter displays how much current is going into the battery and not the amount the alternator is putting out. If the fusible link/connection is bad, the alternator output will appear normal because it isn't discharging or charging the battery. DO NOT CONFUSE A FUSIBLE LINK WITH A REGULAR WIRE. It is designed to be a type of overload protection, quite literally a fuse. If you are not worried about OEM appearance you can get a roll from Napa for about $10 that would probably last you a long long time. If you want OEM, while you order one, use the generic NAPA one to replace the burnt one for the short term. A fusible link will often look OK when it burns but will be crispy feeling and often stretchy. It is the wire from the bulkhead connector going to the battery. It will usually have a rubber tag on it that says Fusible link on it.

#3 Get your alternator tested just to be sure it's OK.

Now for some of the checks you need to look into.

Inspect the wiring for melted wires. Pay particular attention to the large wires going to the alternator, under the dash behind the fusebox and connected to the ammeter. Check the bulkhead connections and the ignition switch connections for burnt degraded connections. Again paying special attention to the large wires.

My biggest cause for concern is the burn wire to the ballast resister you described. There are 3 wires to the ballast resister. One from the crank/start circuit, one to the coil and the other is the run circuit wire. For those wires to get burnt it could be a sign of a bigger problem or it could be a coil shorting out so check the coil to be sure it's in resistance specs.

If inspections leave you warm and fuzzy about the condition of the wiring harness then check for good connections before you install the new battery. Use a battery powered circuit tester or an ohm meter to check the connection from the main alternator wire (disconnected from the alternator while testing) and the battery terminal (No battery connected while testing).
Check the blue and green wires from the voltage regulator to the alternator (both disconnected). The green wire should get connection at both ends and not be shorted to ground. The blue wire should get connection at both ends and connection to the battery when the ignition is in the run position.

Wiggle the wires while checking to be sure they are not getting intermittent connections.

Next check the horn switch for connection to ground. An ohm meter may be better than a test light here because it can check for even small amount of connection. If you suspect the switch then disconnect it for now.
After you get the battery back in and connected. Connect the horns. If they honk without a button pull the horn relay on the fuse box and check it for problems. If they honk and the relay is pulled the problem is in the wiring.

The wires to the horn run along side the main alternator wire inside the engine harness IIRC. If the harness is starting to melt/degrade the horns may be getting the power from there but they would never turn off or would honk constantly while running. If so you need to start unwrapping the wiring under the hood and inspect it.

Let us know what you find.

Here are the basic wiring diagrams for your car They are important when you look for problems.