Sorry, my fault. The cron.log is very long and that's from the beginning of the file which is probably from long ago. I'll post more recent stuff in a bit.
There are no errors in cron.log I went through the whole log. There are a bunch of repeated messages about mailq being empty and others, but nothing about errors. The Warnings I posted earlier are at the top of the file which is from when the server was first installed a year ago so they won't apply any more. Sorry for the confusion earlier.
Just as an FYI, since we couldn't find a resolution to this problem, I decided to try adding a new entry in cron to rm the lock file manually after the server.sh script was run. This seems to work. My system updates properly, everything works and my System Monitor is showing correct information. I'm sure this is not the right way to fix this issue, but just having the issue in the first place was causing major headaches over here, so I had to do something.