You won't need a birth certificate but they will very likely ask the child his/her age. They do it very nicely and in a fun way but when we traveled with my nephew they would stop to give him a sticker and say things like "Hi buddy, having a good time, and how old are you big guy? Always respectful and fun but always asking the child his age not the parents. Unless your child is extremely shy most will pipe up with their age as they are very proud of being 3,4 and 5 etc.
It didn't happen every time but often enough so that if my nephew wasn't still two (three months away from three) and we didn't have a ticket for him we would have by the end of the first day.
They also asked my 8yo a couple times how old she was which surprised me because she is tiny and people often mistake her for 5 or 6 until they get to know her.
Just - my experience, hth
TJ