Does running a cron job of a user require the user to be logged in?

Suppose user 'asdf' is not logged into server 'bbbb', but the server is up. User 'asdf' has cron job. Will it be executed?

Yes it will.
Basicley, my secure practice is even put /bin/false as shell of the designated cron user, so you won't be able to log in or anything and you can modify his cron via crontab command as root.