Can I ask for a certain book edition?
Yes, if you want for example the paperback or the hardcover version of something you can always point out that you want that edition in the comment that you can usually add to your wishlist items! If you don't specify it's up to the buyer what edition they give you.
Can I give/wish for something else besides books?
Yes. You can wish for anything book related - so book marks, book ends, gift cards for online bookstores, book cover protection etc. besides actual books. A good idea is to check Etsy for things here!
Can people from every country participate?
Yes. If you are worried about shipping rates please look them up at the store you wish to send stuff from - it's all listed there. BookDepository has free shipping to a large number of countries all over the world, so a good idea is to look there if you want to keep the shipping to a minimum.
Can I send used books?
Contact the recipient first - if the recipient says yes, then you can!
If I buy gifts through a page like BookDepository, do I have to ask for someone to give out their address?
Yes. If the page you choose to buy the book from does not have an address linked to a wishlist on that site (like Amazon enables) you have to ask the receiver for his or her address - and it's up to that person if he or she wants to give it out! Remember that many might choose to use Amazon because of that the address is protected.
When do we start sending out stuff?
That is entirely up to you. I've made the 11th of December a sign up deadline because things sent before that should at least reach their recipients before New Year's - hopefully even before Christmas.