98 SL1, reversed jumper cables, now I need help!
 

About 2 weeks ago, I left my headlights on and the battery died. No biggie.

I asked my dad to give me a jump. I plug the cables in on my end, he plugs his on his end. Next thing I know, the car isn't turning on, and smoking is starting to shoot out of the negative terminals on both batteries. He reversed the cables in his car (which is fine btw, none of this damaged his Lincoln).

I unplug the EXTREMELY hot (and slightly melted) cables and we wait a few minutes and do it properly. At this point, the door chime is faint and the radio is dead (even though the EQ lights are still on...?). The car turned on, and all seemed to be fine.

A few days ago, the battery light was flickering a bunch, and today the car finally gave up. All the dials weren't working at all, frozen in place, and none of the lights would go on. The wipers would strain to move for a second and then just died.

I replaced any busted fuse, but I haven't gone through to replace any relays or anything else because until today I didn't think I had a serious problem on my hands.

So, is the reversed cable situation why I'm in this fiasco, or could this all be something else?

Also, hello! I'm new here. :)

Im not too sure about the reversed jumper cables. But electricity can do some crazy things to electronics if not used properly. First thing I would do is find someone with a good battery and physically put it in your car and see if it turns over. It sounds like you could have a bad battery or an alternator issue. I would start from there.

Ayup -- friendly charging system/battery checkup at the local AutoZone is in order ...

I removed the battery but when I went to remove the alternator, I discovered a burned/blown wire (maybe 12 or 10 gauge?) running from the starter to the alternator. Would this be that lovely fusible link?

Yes, I think you found it. Not even close to the first car to have a fusible link, and not the last.

Makes sense considering the purpose it serves.

Now, I ask this because I literally could find no info on it through this website or google, and I don't want to attack this until I have all the proper tools.

How do I change it?

You simply cut it out and solder in a new one. I recommend using heat shrink over the solder joints too. Electrical tape won't seal it like heat shrink.