IPhone 5 won't boot up, error 4005 when trying to restore!

Question

First things first - English is not my first language so please excuse the errors I make. If anything is unclear/badly explained just say so and I will do my best to try again.
I am the (right now not so) happy owner of a iPhone 5 64GB since late april. It was purchased online in the germal localization of apple store.
For a month or two everything was alright, but suddenly the phone went out and restarted - at first I panicked a bit, but after a few weeks without any problems I thought that was a one time occurrence. Then it happend again. And again. And again. Sometimes it didn't restart for over an hour. (This was especially aggravating while I travelled and all my tickets were on that phone.) This all was while iOS 6.X was still a thing.
I updated to iOS 7 as soon as it was out and it worked well for some weeks. Then it broke down again and didn't boot at all, it was looped in the process. I panicked and googled and restored it successfully via restore mode. (Against all advice from backup though.)
It worked for a while but a few days ago it broke down again. Annoyed I wanted to repeat it, but restore mode wouldn't work anymore! That "connect to iTunes" symbol won't even show up and iTunes doesn't recognize the phone.
I tried DFU mode, iTunes now connected and initiated the restore process. But while it showed "Waiting for iPhone" the phone (in DFU mode!) suddenly tried to restart over and over (logo showing and fading and showing again...). Everytime I got "Error 4005". (That seems to be a error usually occurring when a cable or an adapter is broken, so I believe that the rebooting of the phone "broke" the connection to iTunes or something?)
What I tried so far:
Switching cables
Crying
Different computers
my main computer (Win7 64 Pro), with my full iTunes library
a new account on my main computer, iTunes without a linked ID
my MacBook Air (OSX Lion) with a linked iTunes and, after reinstalling, a "blank" one
another computer with Win7 64 Pro and a virginal iTunes (this got Error 4014 the first time, after that also 4005)
"Repaired" Apple Mobile Device Support
Crying some more
I haven't had a phone for several days now and I'm giving up. I will contact support tomorrow.
But is there ANYONE who knows this problem? I know that the reboot loop is notorious, but that even DFU mode (and yes, I used it correctly) doesn't work? Hardware defect? (I was VERY gentle with the phone...)
Any help, information and last straws would be very much appreciated. I still hope I simply overlooked something obvious.
Thank you!

Answer 1

Yesterday (April 14, 2014), my 32Gb iPhone 5S running ios 7.1 unexpectantly shutdown. After plugging it in to the MacBook that I normally sync to, iTunes told me that the iPhone was in Restore mode. After some seaching on the web, I decided to let iTunes proceed with what it claimed was the only course; a restore. It claimed that all data would be erased. I went ahead since I had a backup (old). The MacBook didn't have a copy of the most recent iOS (7.1), so it downloaded v7.1, verified it and started working with the iPhone. After a minute (roughly) the iPhone lit up, but didn't proceed past the Apple logo. iTunes eventually showed an error saying the operation was unsuccessful; ending with (4005). Apple support suggested I turn off "Find My iPhone" from iCloud. Tried that, no change in the result. I tried the restore procedure again, but this time after the restore error appeared, I dismissed the error and left the iPhone plugged in to the Mac. After a few more minutes the iPhone continued to start up! I was able to backup all data locally to my Mac, and then complete the Restore. A fresh install of iOS 7.1 and a restore from backup gave me my phone back. In summary, don't give up if you see error (4005). Hope this helps someone. And back up your phone regularly!