i'm having the same problem: battery level stuck at 80%, network set to 3G but not actually connecting, and the inability to charge the battery when it is in the phone - even though it says "charging" the percentage kept dropping (tried different chargers, cables, batteries, and even wireless charger - i can only charge using a separate charger). the battery not charging started about a week ago. the 80%/3G issue started this morning.
no root or any other mods, very few apps installed. i've had the phone about 6 months. verizon. 13B. 5.1.
i did wake up the other day to a frozen "firmware update" screen, but it has not come back, and i've checked for updates (to 6.0 maybe) with no success. this "firmware" screen was after the battery stopped charging, but before the 80% problem. i had to pull the battery after it sat there a few hours (it said "don't disconnect USB" even though USB was never connected, and there was a progress bar at 0%).
UPDATE: this thing may be falling apart in front of my eyes. now my service is disabled and I have no network connection showing up at all.