It sounds like a contact or two may be touching something it's not supposed to, make sure you secured all shields and plastic pieces correctly. While the step by step guides on ifixit are great, sometimes they leave out small details that may not occur to you. watch a disassembly video with no cuts in it online and try to find where you made your mistake.
I think you'll probably have to buy a new part, ipads generally don't have too good of a reputation when it comes to antenna life. I don't think they are sold here on ifixit, but i'm sure you can find them on eBay for around $5-$15. Make sure you buy a quality part, otherwise you may find yourself buying a new part in a few months.
Have you been placing your phone on the computer? I have heard that magnets in the phone can trigger the shutdown. On my 17in 2011 macbook pro I set my phone on the bottom right corner and the computer goes to sleep, this is because there are magnets in the monitor and the computer detects the monitor's magnets and goes to sleep. This is how the computer detects the lid being closed. My friend once did this by accident somewhere else on his computer and it shut down. Make sure you are keeping magnets away from your computer.
This probably means that the screen or the screen connectors have been water damaged. Luckily for you, it sounds like most of the phone is undamaged. Clean the connectors on the board and on the LCD with a toothbrush and isopropyl alcohol. If this doesn't work and the board's connectors appear undamaged, buy a new lcd. Remember to take all precautions when performing this task, ESD is very likely when you are moving your hand back an forth, as you would when using the toothbrush. Remember to ground yourself
Try borrowing a friends battery to test out. If their battery charges then the phone may be detecting that the chemical reaction in your battery may be occurring too slow, and the phone may think that the battery is too cold. If your phone still says that it's too cold there may be connections that cant handle the electricity intake and make be bottle-necking the charge, you may need a whole board altogether. But don't take my word for it, i work on IOS devices more often, I'm not too familiar with the anatomy of the Galaxy S
Have you tried using a different charging cable? Occasionally, when mine become damaged, they act oddly, once even making the digitizer not respond, but I plugged it in with a different cable and all was well.