I had pretty much the same thing happen to me yesterday when mine arrived. It came pre-charged, so I booted it, installed Legacy 4.2.2, then played with it for a few hours.
All of a sudden I got a pop-up saying to connect my charger (even though the battery gauge said I was still a little bit above 50%) and my unit powered off (all in a matter of a few seconds.)
In my case, I didn't try to turn my unit back on, I just connected the charger and went to bed.
Today I turned it on, played with it for a number of hours, it eventually went way below 50%... I opted to charge it around the 15-20% mark, no auto-shutdown this time.
So, I'm not really sure what the deal was... clearly the battery gauge was off, but it seems to be working better now (not conclusive since it's only been once, but it didn't shutdown at 50% again.)
As for your device not turning back on at first, that's actually normal. A device with a dead battery won't turn on until the battery has charged to a certain degree, even if plugged into a wall. All my tablets do the same thing. So, I wouldn't worry about that.
Anyone have any theories on the battery weirdness?