What happens when you gift a city with your UB in it? Do they get a non-ub equivalent? or is it destroyed?

Never tried, but my guess is - non UB equivalent.

When you take a city with a UB, it becomes the Non UB equivalent (Or a normale Building becomes your UB). I expect gifting work along the same lines...

On a related question... What happens if you gift your UU ?

Oh man, I wish you could get other UBs.....

Unfortunately no.

The real good question is, if you gift units with a promotion available, does the AI get the option to upgrade it, or is it just too stupid to know how to upgrade it? I am still debating on this, as I have yet to see my vassals upgrade my gifted units awaiting 3 promotions!

So, my solution is to promote your units BEFORE you gift them. That seems to work.

Gifting UU's they remain the UU. In a team game this is very handy as you essentially have as many UU's as you have players.. Oh and they definatly stil have the promotions available(just try in a team game).