Why Apple? Why?

Did you enable wifi-sync through itunes when your phone was plugged into it.

 

It's really important to ascertain if an issue you're seeing is an actual bug that everyone has to deal with, or some weirdness specific to your device. In this case, your phone powering down at 20% is absolutely not normal behavior. It's *possible* it's a software issue, and it might be worth doing a full restore of your device to see if that fixes it. It sounds more like a physical problem with your battery, and I would take it to an Apple Store ASAP.

 

I have no problem with WiFi sync between my iTunes and my iPhone 6 plus.
I previously used wifi sync on iPhone 5s and Windows 8.1 machine, and iPhone 6 plus and a macbook air. Works on Asus router and on airport express.
This is definitely not the norm if you can't wifi sync, could be your set up at home.

I'd be interested to see the screenshot of the Settings > General > Usage > Battery Usage, and the background app refresh screenshot too while you're at it. Let's see what's eating up your battery.
You could also have bad battery.

 

I've seen people complain that their battery percentage is a little jittery, but nothing like this. Turning off at 20% is kind of a big deal. If it is a bug it's certainly not a consistent or easily reproducible one across different devices.

 

Personally I only used Wi-Fi sync once or twice and stopped because I found it to be incredibly slow. I could go get coffee, use the bathroom, make breakfast, shoot and edit a 15 minute movie with FCPX and use Compressor to export it for YouTube streaming and my phone may have just finished. (Both my computer and phone use AC networking with my AC APEx).

Anyway back on track here, sounds like you may have a defective battery. I'd take it to an Apple Store, AASP, or call AppleCare (800-APL-CARE) to get it diagnosed and repaired. Your phone is under a year old and if bought from an authorized reseller you should have no problems.