I do have all options in Desktop Search disabled and I still have a multiple akonadi_agent_launcher processes running and consuming all my CPU for several minutes. When I change settings in .config/akonadi/akonadiserverrc
to StartServer=false
and restart, it stops but I can't use Kmail, Kalarm. Is there any solution?

you could try https://launchpad.net/~pali/+archive/kdepim-noakonadi

KAlarm can be built without Akonadi support, IIRC. Recent KMail versions cannot.

I will eventually try kdepim-noakonadi, but tell me please, is it normal that several (about 20) akonadi_agen_launcher processes are running simultaneously, even when all options in setting are disabled and I even unchecked all folders that were marked to be indexed? I am not sure if I understand the options. I thought that when there is no folder marked to be indexed nothing is to be indexed and akondadi may be still available for applications to start it but it should not index anything. Looks like it doesn't work that way.

Akonadi is not an indexer. "Agents" are programs that do specific tasks, like retrieving mails from IMAP servers, handling calendars, sending mails, and so on.

I think my problems resulted from a bug
https://bugs.kde.org/show_bug.cgi?id=302532
Similar issue was reported in topic
viewtopic.php?f=63&t=99137

I followed the workaround from above linked bug and deleted all enties in System settings → Personal information (where Akonadi configuration is) and restarted akonadi and now it is working ok.