The notion of having no state is kind of an impracticality, do we accept that as a given?
All pregnant women are moved to Region0. They give birth and then moved back to their place of origin right after the birth. The babies stay there and are raised for 18+5 years (pre-school, school, Uni) and then they audition for their 10 countries of choice (kKinda like the 'draft' in the NBA)
That sounds like a fine plan to me. "Auditioning" could consist of military or other volunteer service.
> At what age?
18, unless you're an orphan or emancipated, in which case you can do it sooner. Before that you inherit the citizenship of your parents or legal guardians, but only until you turn 18. Then you have to choose. Most people would choose to retain the citizenship they grew up with, but it would have to be an affirmative decision. I don't think it's right to be a citizen of a country to which you have not pledged allegiance as an adult.