IPhone 4 :: Reboots / Shuts Down Even When Plugged In

Here are the specs: iPhone 4, iOS 7.0.4, 16gb, GSMÂ

I've never had any big problems with this phone while using it, but today it randomly shut down, so I thought it might be a battery information glitch, because it has happened with a different iPhone of mine. So, I connected my phone to electricity, it turned on and stayed like that for a while (I had fully charged the phone a few hours ago) and it showed 94% of battery left, which is quite normal. The odd thing was, that it was searching for carrier and it was impossible to turn wi-fi on, the button was not responding.

So I went ahead and rebooted the phone, still had the same issue. While still connected to the charger, the phone suddenly died again, showed low battery and turned on again. After a minute or two, the same thing happened again and after another reboot - again, again and again. I tried to reset the settings, still no carrier and still rebooting, even while charging. I tried completely erasing data and settings, but it was impossible to do that, because it asks for the iCloud password. Since it cannot be connected to the internet, there is no way for me to do that. When unplugging the phone, it dies in the same second. Next, I tried connecting the phone to my computer, but I can't even get it to turn on for a minute, because it reboots all the time after a few seconds (doesn't get enough power through the usb, I guess). Tried DFU mode, still no success. While connected to the computer, iTunes won't show the device connected to the computer.Â

So, there is no way to update the OS through the phone, because there is no way I can connect it to the web, resetting the settings won't do a thing, completely erasing the phone is impossible, because it cannot be connected to the internet, impossible to connect to a computer, because it reboots all the time, so no way to restore the phone.

Info:
iPhone 4, iOS 7.0.4