It sounds as though you have a battery cable problem at the positive terminal: if it gets hot, you've clearly got resistance at the cable, which can prevent a start and mimic a dead battery.

As for the reversed polarity, I think you are OK if you did not start the car. The initial sparking when you connected the terminals incorrectly was likely due to lamps and low-grade current supply. If you tried to start the car, I have no idea what reversed polarity would do to the ECUs and other digital electronics. I also doubt that it affected the alternator diodes, but even if these were fried, you could replace the regulator for only $50 or so.