Happened to me when I bought a bad LCD from China, it started doing this when I bent the ribbon cable to put the screen back on the place. The seller send me a new one, it’s probably related to the ribbon cable. I don’t think you can fix it yourself other way than buying a new one.
Happened to me when I fixed my friend's iPhone, for me it was only a bubble of humidity that was stuck between the metal plate and the LCD, I opened the phone, removed the metal plate, let the phone open for 5 min to be sure that the humidity was gone (you can use a hairdryer if you are lazy), and put everything back.
You can try a lot of things. Replace the Wi-Fi antenna. Try a Hard Reset (home button + Power Button for 10 Sec). Sometime restoring the phone works. The iPhone 4 antenna isn't really powerful, sometimes you don't get any signal only because your hand blocks it, maybe you put something that block the signal inside the phone.
Maybe a battery expert would have a better answer, but with all the battery problems we see on the news, I wouldn't take any chances to burn myself or broke my phone. A defective battery is really dangerous and can blow up at anytime.