iPhone 5 shutting off and on with iOS 7.0.2

Today I filming with my iPhone 5 and it was at 78% and it just tuned off and I got the pictures that says please connect to charger and then I waited and was able to turn it back on but then it showed 18% I turned it off a few times during the day and even left it off and when I turned it back on it did not change so then it was at 10% and stayed there all day at 10% and now i just turned it off and back on and it showed 28%. So now im going to let it die completely and then restore my iPhone then charge it to see if that works.

It also turned off and on before I did the update to 7.0.2

also, does charging your iphone with an ipad charger hurt the phone? I'm in France for school and I got the apple travel adapters.

one more thing, if your charger is not plugged in all the way in the iphone (it is charging though) will that hurt the phone? I made a dock out of wood and I can not see if it is all the way connected so I'm not sure that will change anything. It is charging but I still worry.

Please help!