What is happening?
When something (chip or other) is shorted on the logic board, it can cause power to be rushing to ground, and consuming power from the battery very fast and causing overheating of the board. Thus causing some or all of the above.
Now for the iPhone 5:
The iPhone 5 was the first iPhone to which Apple introduced a chip infamously known as U2 (google it).
It is responsible for telling the phone what is plugged into it (car charger, wall charger, computer, etc.) and consequently what the phone should do next (start charging from wall charger, start communicating with computer, etc.).
U2 gets easily damaged on the iPhone 5 and can cause ALL the symptoms you are talking about. It is the most likely culprit in this case, although as said first, anything shorting on the board can have similar consequences.
This signature failure is not easy or cheap to remedy because the chip is located right next to the CPU which can very easily be damaged (actually the solder joints underneath it) from the heat required to replace U2.
So if you are sure your battery is not causing fast drain and heating on another known good phone, then your issue is as described above.