Sounds like possibly a battery issue, if not then a logic board issue. Try a new battery first, if that doesn’t work then it’s more likely a board issue.
This would be an LCD issue. Only physical damage could have caused this. Whether it was dropped, or hit, physical damage is the cause. Pro model iPads LCD replacements are very expensive, especially the bigger models. Cost alone on the part can exceed $200.
How did this originally happen? Was it dropped or damaged in any way? This sounds like a common battery issue but any extra amount of info you can provide would help!
Those are gold contact points for the rear cover of the phone, not sure what they are used for, as the fingerprint sensor is not connected to that in any way.
That should fix the charging/cable issues, yes. If you decide to attempt this repair, just make sure to be careful, take things slow and try your best to put the new part in the exact place as it was from where you removed it. This part has contact points and an antenna cable which can be finnicky with placement. So if it still doesn’t work after your first attempt, try re attaching the antenna cable, making sure the gold contacts make…well…contact and making sure the actual cable is seated in it’s female counterpart correctly!
The flex cable on the inside might have been damaged or unsocketed somehow. You should be able to open it up and double check. If you aren’t comfortable with that, you could try taking it to a local repair shop if there are any in your area!
First thing to test is if the device even receives power. Do you still have the old screen? If you do, see if you can plug in the old screen and see if the device turns on still or even receives power. Other option is if you have a USB ampmeter. Just gotta make sure the battery is plugged in. Let me know how this goes!