Windows 10 - cannot open any Windows app, settings, updates

I've been using Windows 10 for something like 2 years already and everything was fine, but recently, around 3-4 weeks ago I noticed I couldn't open Calculator app (the pre-installed one from the Store). I clicked on its icon in Start menu and simply nothing happened.

Then I started investigating and searching. I found some articles on restoring Windows Store apps using some commands in PowerShell (described for instance here). I got many errors during this fix trial:

and it didn't help.
Currently I cannot even open any "system place" from start menu like "Settings", "Troubleshooting" - I can find it in start menu, but nothing happens when clicking on it.

I've also tried this this Microsoft's troubleshooter which said it cleaned some cache of windows apps and the Store opened, but I still cannot access any apps like before.
Also when I tried to install some app in the Store, the download was started and was very slow (~100kb/s) and it closed itself after several seconds.

I'm also not able to open Windows Updates - when I find it in start menu and click "Check for updates..." nothing happens.

I scanned my PC with Avira and Malwarebytes - it found nothing.

Any help would be appreciated.

Answer

Firstly, create new user account on your PC and see if there is the same issue.

If not, repair your corrupted user profile:

http://windowsreport.com/corrupt-user-profile-windows-10/

Otherwise, it could be caused by system components.

Please try to repair your PC without losing anything:

How to Do a Repair Install of Windows 10 with an In-place Upgrade
https://www.tenforums.com/tutorials/16397-repair-install-windows-10-place-upgrade.html