Here’s what happened. The load increased, clogged CPU time and my account was suspended during my night for overusing system resources.
It took me 10 hours to get my sites back. Tickets did not help, but the live chat did the trick.
I still believe that the problem #1 is MySQL database. MySQL locks up, PHP processes spin trying to get the info (I think problem #2 is PHP). However, nobody was able to find the mysql logs to prove me right or wrong.
It’s much easier to just disable the account, than to have a look at what/where/why the problem is, or simply kill the CPU/RAM hungry PHP processes.
The provider wants me to migrate to the “Business class” (5 times more expensive). I’m seriously thinking about that…