Stuck at Unknown error 4005 boot looping

Hi there!

Think I got a hard one for you guys here:

My dad have a iPhone 5 (iOS 7) that one day where laying out in the sun, and he was getting the "to hot" iPhone error. He cooled it down and it worked fine. But after some time, the iPhone started to randomly power off and restart in safe mode. He had to take a complete reboot to get it restarting. This usually happened when he had 20-30% battery power left. He wanted me to replace the battery, and so I did. (told him that I was pretty sure this was a software issue (have tried with restore) Anyway, the battery replacement did not help anything, so he just went along and bought a new 5S.

The iPhone have been laying alone for a month now, and I was thinking on to see if i could fix this now. Tried to start with a restore, to then not restore from any backup. Then my problem started:

Now I´m stuck in recovery mode (Connect ti iTunes logo) and there is no way i can get back to the home screen. I only get "unknown error 4005" from iTunes, when trying to do a restore, in the "waiting for iPhone" process. The picture on the iPhone freezes. Have tried these things:

DFU mode

Recovery Mode

Disconnect front facing camera

Disconnected the battery (resulted in error 21)

Restore on two different win7 64 bit computers and one mac

Repair iPhone Device driver in win7

Uninstalled and reinstalled iTunes

Have NOT tried with tiny umbrella to kick it out of recovery mode yet.

The home button and power button works fine, since i can get in DFU mode and turn it on and off. iPhone stuck in Recovery mode after Battery replacement..

Any tip? Replace the lightning port? Could also try with a original battery, but cannot see how that could make this error.

Maybe a little messy thread, sorry for that! Thanks!