Use rock jugs. It's much simpler and faster and requires no fuel. As for glazed/unglazed jugs in stockpiles, I can't speak to this as I have yet to create any ceramics. If you must make things out of ceramics, make crafts or statues, or hives for that matter.
Your stockpile settings look fine. I would switch the tools stockpile to accept all materials. (And switch to rock jugs).
You extract stockpile is empty as they are all going to your tools stockpile instead. Try setting the extract pile to take from the tools pile to free up room there for empty jugs.
Set all hives to install and gather -- i.e. build them and leave them. Personally, I can see no good reason to tamper with these settings. I tried it in a previous fort and to put it simply it was a pain in the ass, I saw no noticeable increase in production, and I also encountered inexplicable problems. In fact, this may be the cause of your problems.
If you still have problems after switching all hives to install and gather and ensuring that you have lots of empty jugs, deconstruct and rebuild all your hives. (Maybe don't do them all at once, but in batches as if starting from scratch.) Hopefully you won't need to.
Yes, mead needs barrels or pots (rock pots rule).
Royal jelly can be eaten raw or cooked, however, I believe cooks view it as an ingredient of last resort.