Sounds like you may have some problems with wiring that has rubbed against the body some place to wear through the insulation, causing a short. Or perhaps previous owner did some wiring of his own to add an accessory and messed it up...
Start with the melted fuse box and make sure all the fuses are the right rating. If fuses are the right size, you can still get a hot spot in the fuse box from resistance due to dirty connection. Had that happen in my son's car. At least the fuse box will need to be replaced. Once that's squared away you can see what's under the dash at the heater controls.
For the door locks, check the archives--I know it's come up before, but can't remember the specifics, perhaps a bad ground.