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.

Possibly. Have you tried talking to the carrier it is currently locked to?

If you're on the fence about your abilities, give this a try. It really isn't too bad.

I decided to remove the old (shattered) digitizer with an Xacto knife as I had seen elsewhere. That made quick work of that stage, but I would recommend gloves and being extra careful on the sides. The rest is fairly familiar, a lot like an iPhone if you've ever repaired one of those.

Another tip is to use a magnet from your fridge to keep the right screws together.