To clarify this even more: Your problem is not mails not getting through (as your test emails show). Your problem is that your install is quite busy, and will take some time for these mails to be sent. Wait until the enqueued count has dropped to 0, or try to speed things up as @gerhard explained above (it will still take some time, though).
Until then, it’s normal that a lot of things in Discourse will appear broken. Also, don’t be surprised by all he requested mails arriving when the queue clears – they’re all in there and will be processed when they get through.