I have an original Apple Smart Case and never experienced this. Than i cracked my screen and went for a replacement. Now I have ghost touches and this magnet thing. The question I have is the guy changed the digitizer glass so what can he have moved to do this?
If the phone has worked after the hit it should work again. The screen is gone for sure! The question is what else. You say the phone is 4 moths old and I think it is best to send it to an Apple store to swap it! You will pay a little but you will still be covered afterwards for other defects!
A lot! Wireless and GPS too will have less and less Signal. I lost that too due to a non authorized service point (the guys who think that the producer has put that part for fun there) and I ended up taking the phone to Apple for exchange! Find the clip and put it in it's place if you want to have an operating phone!
I would suggest first to try another charger! Than by what you say, "It had been taking a really long time to charge lately, but once it's charged, the battery seems to hold it just fine.", the battery seems not to be responsible for this situation! I would suggest to take it at a service point since it is out of warranty. If it really has problems with charging circuits (which are docked to the board) you can't do much on your own.
OK Still the same problem on dropping the iPhone 4! The logic board has moved you have to unscrew it (ask the service shop people to do it) and tighten the screws again. The screws grounding the logic board to the metal body are responsible for GPS, WI-FI, Second mice, GSM antena etc.