12.6 on new battery; no start, locks malfunctioning, and possessed gauges

[bodega1027]

Vehicle: 2007 Chevrolet Aveo 1.6L MFI DOHC 4cyl

The Problem: Got sister a new battery when the old one tested dead about three weeks ago. Yesterday the car went dead; no start, no dome lights, and no headlights.

Today, before I could come over, she noticed a clicking coming from the doors that sounded like the lock solenoids repeatedly going off; this only happens when you lock the door, either by the fob or locking the driverside (which triggers the auto lock feature on the other doors). Further, we previously had a trunk that wouldn't unlock, even manually with the key. It now unlocks just fine.

Played with the plunger on the driverside with the key in various positions, and it would trigger new glitches, such as the instrument panel gauges' "wiggling", or a loud clicking coming from the dash, kind of by the fuse panel. Battery read 12.6 volts, but still no starting or headlights.

Tried to start and, it she says she saw a wisp of smoke come off one of the battery terminals. Jumped it and it came on just fine, and ran it for maybe ten minutes with all the accessories off. After shutting it off, the problems began again.

I'm not even sure where to start. Security system? Power lock system? Bad battery? Ground issues?

This is my sister's only car, and she's got a job and a 2-year-old that need her mobile as quick as possible. Any help would be greatly appreciated.

EDIT: Pulled the door lock fuse, jumped the car again and let it run for awhile; all the same problems persist when I shut it off. Took the battery to Autozone and they said it was fine; only 88% charge, but it should be able to start the car. Got it hooked up to a drip charger right now to see if there's any change at 100%, but I don't know what else to try.