It appears to me that you have some misunderstandings about おいくつ...
Is it natural to say 「おいくつですか」 to children who are around 10, or younger?
No, it isn't. おいくつ is an honorific expression. It's a respectful and formal way of saying 何歳 or いくつ, and it's funny to say おいくつですか to a small child. Usually "いくつ?" or "何歳?" is enough to a small child. On the other hand, saying 何歳(ですか) to a mature adult can sound blunt and impolite, but it largely depends on the relationship between the listener and you.
About what age to stop using おいくつですか？
Provided you have a good reason to ask one's age, you do not have to stop using おいくつですか even if the listener is over 100. Instead, you should start using おいくつですか when the listener is old enough. (Of course there are even politer expressions like お歳を伺ってもよろしいでしょうか.)
About 10 years old is the cut-off age for asking another person's age?
Asking the age of an adult for no good reason can be rude by itself, but that's a sociological problem which is out of the scope of this site. (Well, talking about age is not recommended, but it's not a taboo, either. I know the rough age of all the friends and colleagues of mine.)
And the clever / funny way to ask is 「おいくつですか。」
おいくつですか is not funny at all if used in an appropriate situation. What made you think it's funny? "Clever" is not an appropriate adjective, either; it's just a basic honorific word every adult should be able to use fluently. A "clever" way of knowing one's age is like talking about their favorite game/manga/anime in childhood :)