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Answer to "Won't turn on without cable, won't charge."It could be the logic board. It would explain the overheating, unrecognized battery, et cetera. For the battery, the one fix I've seen before is reseating the battery connector on the logic board. That was when a battery was not charging at all, though.
Answer to "Can I use this phone on a verizon wireless network"Possibly. Have you tried talking to the carrier it is currently locked to?