The exim logs are /var/log/exim_*. You'll want to sift through /var/log/exim_mainlog* and /var/log/exim_rejectlog*
There are many functions in Cpanel Dedicated Server which can be managed through ssh.
When cPanel & WHM is installed, a number of predefined scripts become available in your server’s /usr/local/cpanel/scripts directory. You can use any of the scripts simply by typing its location in a terminal session. For example, if you wanted to run the update script, you would type:
Here is a list of links which I have come across recently to secure cpanel servers.
From the Official Documentation