you're sure this is iPad 3 and not iPad 4?
you were on the right tracking thinking it could be charge port. This is unusual in iPad 3 but common in iPad 4 to have damage to the usb logic ic chip due to non-Apple charger damage.
I don't have any other suggestions for you, it is likely a hardware defect. You could google the difference between recovery mode and DFU mode just to make sure that you are in true DFU mode before attempting to restore via your new charge port.
jessa