No. Both vehicles have voltage regulators as part of their charging systems to prevent overcharging the battery. You CAN easily overcharge a battery with a home battery charger. but with jumper cables, it's just as though your battery was bolted into that other vehicle. The only exception is if that other vehicle had a shorted voltage regulator but then that owner would have been having problems of their own.

If your battery melted to anything it is more likely there's a problem with your car. That overheating and melting doesn't happen quickly. It will take many hours or days. You didn't say what "broke down" means but it suggests there is some other underlying problem that hasn't been diagnosed or addressed yet.

okay well he also said that the starter had gone bad and one of the 4 cylinders. could that have caused it?

Parts fail all the time. There's a lot more to the story that I don't know. As for the engine, a lot of things can happen to one cylinder. Does he mean it's misfiring? That is due to a bad spark plug, spark plug wire, fuel injector, a valve issue, or something like that. Those kinds of things are relatively inexpensive to repair. He could mean the cylinder wall and / or piston rings are worn or damaged. Those are mechanical issues that are expensive to repair.