So I'm not sure what is going on, but the aforementioned item isn't being marked as complete... despite me learning the recipe AND the item. Help!

You are more likely referring to the recipe:

Go to the first tab of settings and choose:

Then open up your profession window as this allows Blizzard to send us information for your professions.

I'm trying to do that but its not actually updating when I open Engineering for some reason.

I get a weird window flashing next to my profession window but then it disappears instantly.

Recipes recently went through a much needed refactor. This issue is likely resolved by now. If not, please reopen the ticket.