This is so strange. In order for this to be happening something would have to be interrupting our script after the order is completed but before the order confirmation email is sent out. That is a very narrow window to interfere.
We should test for conflicts in light of having tried most everything else. Basically what we want to do is deactivate all plugins except The Events Calendar and Event Tickets (Plus) and switch to the default Twenty Sixteen theme. Then try “purchasing” a free ticket. Did you get any email regarding this? I could definitely see a couple of your plugins that might be causing the issue, but the theme is also suspect as a malfunctioning override could be causing part of this.
If the above does not work, would you be willing to upload a copy of your database for us to try running some tests on?
Cheers!
– Brook