This may not be what you want to hear, but can you take it to an Apple Store and get a replacement? Seems like you're still in your warranty period. After performing the hard reset and reinstalling iOS, if it still doesn't work there may be no hope, except Apple. Good luck. Let us know what you come up with....
I had this problem with a 4s I had last year. I got in touch with apple and they asked if I had used any non apple charging devices and if so should try using the one that came with the phone. Well as it happens, I had been using a mophie copy charging case. So I took it off and used the apple charger and it did the trick.
If you have used any non apple chargers this may be your problem.
Yes, you will have to tell them the previous problems of your phone, why did you change it anyway? You could have asked Apple to fit a new battery for you, or you could have requested a replacement phone.