stuck in a loop where it flashes the apple screen

Hi, I have a totally straight, never jail broken iphone5. My battery was on its way out and this morning when I came down my phone would not boot from the red charge screen. I promptly went out and got the battery replaced, thinking that would solve the problem. Unfortunately the phone now seems to be stuck in a loop where it flashes the apple screen and then the connect to iTunes icon. I can occasionally get iTunes to recognize it and it asks for a restore. But the restore hangs at "preparing the iPhone for restore" and gives me an error message. SO far I have gotten a failure to restore unknown error 2001, 2006, and 2008. I understand that some of these errors refer to my charging cords, SO I have tried two different cords to avail.. When the phone was given back to me after the battery was replaced, the man was seemingly confused by the looping apple screen... but he did say "these batteries come mostly dead but usually have a bit of charge in them..." I was wondering.. DO you think that the phone is powering down to restore and then doesn't have enough charge to power back up? I also seem to be having trouble turning it OFF or getting it to recognize DFU.. Unfortunately the phone is out of warranty so apple will not help me But I was hoping you could shed some light on what you think is happening. Should I go back to the man who replaced the batery and see if he can put a different one in perhaps? What do you think?