Check all connections and clean it make sure that everything is not damaged or burned then unplug everything and put it all back together and put it on the charger maybe that can solve the issue. Hope this helps.
Check connections it may be heating up because of a poor connection also double check the screen if possible if its for a different model iPod it will have a huge draw for the power making the signal weak. Let me know if this helps please.
Check all of the connections and clean them if possible. there may be a connection issue check and make sure there are no breaks scratch or dents anywhere that seem out of the ordinary and see where the adhesive lies and make sure everything you took off was just adhesive.
The dock connector is the problem check the pins on it and clean it. if there is pins missing it could register the charger but it can't provide a charge because that pins is missing or so forth. My bet is you might have to replace the dock connector if the battery is a confirmed good battery and the charger is a confirmed good charger it has to be the dock connector if its missing pins it can see the charger but not get a charge.
Most likely its the phone if it worked on the galaxy and not the iPhone it is either non compatible or the phone is stolen or lost and registered as such which there is no way of bypassing. So the sim card is either non-compatible or the phone was registered as lost or stolen and when the iPhone updated it realized it was registered as lost or stolen and then shut down its service. But in case it might be a hardware problem, Check the connectors of the sims card for any damage or burns and clean it. Unless you know that it worked before and it stopped after the update in which case it can't be hardware . Hope that helps.
You might have fried the board check and clean all connections but my guess would be the board fried when the electricity jumped off the screw. Also check everything that was connected to the board and the screw that had the electricity because these are also in danger of being fried.
It to me seems to be the video card as the most logical problem it seems to be that the video card is either not picking up the screen or is corrupted itself or the driver for it has been corrupted I recognize this issue vaguely from a different computer I had worked on my bet is that the video card has been corrupted. So you may have to either buy another video card or find a way to check the drivers because one of them may be corrupted as well.