If the engine will not run when you disconnect the battery that means your alternator in not supplying enough voltage, that doesn't mean your alternator is bad it means the engines electrical system is not receiving enough voltage, now think about the melted negative terminal, the battery was supplying all or most of the needed voltage, if that neg terminal was at all loose it could have caused this due to arching. I have to assume your alternator has an internal voltage regulator, usually I see 14.5-ish volts coming from the alternator. I am leaning towards a bad alternator, you can have it checked at any parts store. I do not have a electrical diagram.
There is alot of experience here so keep us updated and you will get the help.

Fran