The batter almost never gets damaged because of a simple drop. So I would rule that out.
When you say, itunes recognizes the phone "occasionally" what do you mean exactly ?
Try disconnecting the screen and connect it to your computer, will it be recognized ? And most importantly if you leave it like this for a few minutes, and gently feel arround with your fingers what exactly is getting hot, or does it only get that hot, when the LCD is connected ?
These questions could lead us to the problem.
if you have a good camera you should try to provide us with a clear picture of the lcd connector on your logicboard.