[PROBLEM]: Menu and soft navigation unresponsive after drop + more

[mokite]

I am trying to help my nephew with his dysfunctional Sony Xperia Z3. Strange things started to happen after he dropped it.

Some of the issues may not be related to the drop. My nephew are eleven years old and only play games, he have no idea if any of these things worked earlier. It is VERY important for him that I backup before a potential reset, as he have some game data and possible (?) local saves that he dont want to lose for anything in the world.

Side note: I am not using Sony myself, so I am unaware of any model specific quirks that I need to pay attention to.

I did spend some hours searching for a solution, but can't find anything useful.

• The display is *NOT* broken, and the phone have no third party screen protection.
• The phone have been restarted, turned off etc. I've tried to boot to safe mode.
• According to the hidden Service menu, the rooting status is (empty), and the bootloader can be unlocked.
• Android version is 6.0.1.
• Model number D6603.

I've done the built-in Sony Xperia tests, and everything seems to work fine from there.

LIST OF PROBLEMS

Issues possibly connected to the drop:

1. The notification/top menu can only be pulled down half ways. Then it gets stuck and have to be pulled up again. It always state "No notifications". None of the icons can be pressed. No difference between portrait or landscape mode.
2. Only the back key in the bottom navigation bar (the soft navigation) works. The middle and the right button do not.
   I don't believe that the touch-screen digitizer is broken, since I can push the buttons and I get both a vibration and visual feedback. The problem is that there is no action at all when I press the buttons.
3. I can't download anything from Google Play. I have a connection, I can surf, but Google Play is stuck on preparing the downloads with the eternal message "Download pending". I can download and install apps from for example ApkPure, so the abobe is probably an issue related to Google Play. I've tried/checked this so far:
   • As far as I can see, there is no download restriction.
   • I've cleared the cache and data of Google Play, Download Manager and Downloads.
   • Mobile connection or Wifi = same problem.
   • I did not log out from Google Play. I don't use Google Play myself, so I am not sure what will be erased if I remove the account. Will everything be restored when logging in, since I cannot backup (see next point)?
4. Similiar to the above problem, I cannot do a backup (it goes halfways, throws an error, and then exit). This is bad, since I can't do a factory reset if I can't do a backup first.

Other strange issues:

• 
  I can't enable the Developer Options. The usual trick with tapping the Build number menu is NOT working. This is the recommended way for this phone model, as far as I understand.
  An article at Sony Developer site (I can't post a link as I am new here), about unlocking the bootloader, implies that an unlocked phone, already will have the Developer Options enabled;
  "On your device, turn on USB debugging by going to Settings > Developer options and click to enable USB debugging."
  This is however NOT the case.
• The phone cannot boot into recovery mode (there is a Sony recovery, right? Else, ignore this...). I am holding the power button and the down volume button while booting up (as per recommendation from the Sony support page), but no go. I also tried with pressing the volume up and down while booting, as recommended on various web pages. Not working.

I would be VERY grateful for any ideas on how to solve this. If I can at least enable USB debugging I can probably work things out from there. Thank you in advance!