Those all look like standard system-generated messages. For example, logwatch is installed on most modern linux servers by default. It looks through all the logs on the machine and sends a nightly email about what's going on on the system.
The logrotate message about conman is just telling you that /var/log/conman.old doesn't exist. From the logrotate man page:
olddir directory
Logs are moved into directory for rotation. The directory must
be on the same physical device as the log file being rotated,
and is assumed to be relative to the directory holding the log
file unless an absolute path name is specified. When this option
is used all old versions of the log end up in directory. This
option may be overriden by the noolddir option.
so just create that directory as root (sudo mkdir /var/log/conman.old) and those messages should stop as logrotate will then be able to rotate old log files into the directory.
You should be able to adjust the WHM backup messages through the cpanel interface to your server.
If you don't want to see the logwatch messages, just move the file /etc/cron.daily/0logwatch out of the cron.daily directory to stop the nightly logwatch processing. I would however encourage you to actually inspect these messages occasionally to keep an eye on what is going on with your server.
These messages all look relatively normal. Once you fix the logrotate error you should be fine.
