Constant MysQL crashes. How to fix?
So I finally fixed it. 

I ran upgrade script as described in question.

The logs complained only about the wrong structure of performance_schema which is a system database that contains only performance metrics and a missing proxies_priv table inside other system database. 

The errors are gone.

And of course before making any changes I did a backup of all databases with mysqldump. 

Thank you all for help! :)