iPhone not charging while dead and hot while running?

I got this iPhone 4 from a friend thinking it was a battery issue when he said that it died and wasn't charging, i decided to investigate.

First thing I did was put my battery into the phone and it worked great except the iPhone was hot on the bottom left (looking from the back) part of the mother board where the battery plugs in and was draining the battery fast. I was playing around with the phone until the battery died, then the same problem happened. It doesn't charge but the apple logo comes up and it turns off (when plugged into the computer) i couldn't restore it or anything even when put into dfu mode and recovery mode. i decided to plug it into the wall but all that happens is it comes up to the ios 6 activation screen says searching for the carrier (but there is no sim card installed) and the battery says 100% (is obviously dead) with the plug in it. It tells me to activate with iTunes but it when i unplug it the phone shuts off.

I put the other battery back in that i charged with my phone and the phone powered on like normal again. I then tested the broken phone if it charges the battery and it does but when any battery is completely dead it wont charge it or give me the dead battery screen iPhone's and iPods usually give.

so if anyone can help me it would be much appreciated, thanks :)