I am unable to turn ON iCloud synchronization - buttons simply freeze

Greetings RomanVybranovskyi,

If you are unable to connect to iCloud , these are the best steps to start with: If you can't connect to the App Store, iTunes Store, or other Apple services - Apple Support

You can also check to see if you have any restrictions enabled, if so this could be why your settings are greyed out. Use parental controls on your child's iPhone, iPad, and iPod touch - Apple Support

We hope this helps!

Sincerely.