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Wilzer the only way when Cron is not actually running is if you configured it and then removed.
If you have installed the software but never configured Cron, you still have Cron running ( every time when anybody refreshes the page. I think interval is once in a minute for that default Cron). Oxwall is set this way because as mentioned by Tammy Cron is a must have for Oxwall and to complete daily routine Oxwall came up with this default Cron that runs on page refresh. It wont' work for larger tasks ( ex profile import or huge mass mailing).
You also could not have destroyed your DB while configuring Cron.
Yes, site opens when Cron is not running.
I did write what Cron does in Oxwall in my post here. Hope that it will help you:
https://skatips.wordpress.com/...oxwall-what-is-cron/ .