Can I use "little" meaning the age?
You are still little to do it. Wait until you are 10.
If yes, are there any situations when "little" is preferable over "young"?
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"Little" also means "young."
(Of a person) young or younger:
my little brother
Similarly, you can call your older siblings "My big/older brother/sister."
Both, "young" and "little," are correct and can be used interchangeably, but I will be more inclined to choose "younger." Though "little" is more commonly used.
Do not say "small brother/sister," unless you are referring to their physical attribute.
You're just a little kid. You don't know anything is a typical thing to say. Little does not just refer to the size of the child, but to his young age and inexperience. Little refers to everything about the kid being a kid, the child's "kidness".
Oxford English Dictionary:
Of a child or animal: not close to being fully grown, (very) young; (also) younger than a particular relative
Thus, we can also refer to a younger sibling as either my little sister/brother or my kid sister/brother, and they mean the same thing.