by Robert Brenstein.
Are you sure that cron is set to run automatically every 20-30 mins? Both, the email not going out and backup not run, could be caused by cron not running on schedule. You can test this by running cron manually like 3-4 times in a row (some cron functions are executed only every so often). If mail goes out and backups are done, you have an issue with cron. Otherwise, you need to investigate each function for the cause of malfunction.