I suspect this is a hardware (firmware) issue, I've had it, along with numerous other people and a replacement followed by an iCloud backup has always fixed the problem.

It's not normal battery drain, nothing could cause the battery to jump from 30% to 0% instantly. It wouldn't explain the percentage jumps when plugging back in, or how usage in the cold causes shutoff when under 50%.

I fear Apple knows there is a defect in millions of 6/6s devices and they are hoping they can ride it out though natural phone attrition.