Ok, I was finally in contact with a Senior Advisor and from their diagnostic they told me that my device passed all the checks with Green Light, which means they would not be allowed to open it to see if something is wrong as the hardware seems to be working perfectly fine.
Now, I performed a System Restore with their assistance and have configured the phone as a new one.
Following their instructions I have not copied the backup.
I am instead downloading the apps once more and inputting the datas inside.
According to the S.Advisor what happened is that I made the system update via a wi-fi connection which was not perfectly stable and this corrupted the downloaded file enough to make the energy readers in the battery trigger a shut-down at around 30%. She advised me to always update the phone via iTunes to avoid almost all useless complications.
Concerning performance, I noticed that choosing a Live Wallpaper for both Lock Screen and Home screen induces an awful lot of lag.
I will keep you up to date during tomorrow to see if the problem is solved.
Only thing: Messages and Health data is lost after update, maybe they were not included in the iCloud datas?
Anyway now I have direct contact with the S.Advisor and she will follow me directly from now on.
Thanks you all, I will copy this same message in the other post I was keeping, hoping this will help somebody having the same problem.