철부지 is very different from the rest. It means immature. So usually it's used to refer to an adult who acts immature. You don't say a kid is 철부지 cuz kids are supposed to be immature.
아동 is pretty formal compared to the other two (어린이 & 아이). I usually see this word incorporated into some legal terms like 미취학아동 (kids who are not attending school yet) or 아동보호법 (law protecting kids) so yeah usually pretty formal. Not really used in every day conversation by itself. At least not as commonly used as the other two.
어린이 and 아이 are interchangeable. But then there are some subtle difference I guess cuz depending on context, one is more appropriate than the other. I feel like 아이 can be used in more personal way. So you can say 우리 아이 to refer to your own kid but you don't say 우리 어린이. Also, you say 어린이집 (kindergarten or preschool) but you don't say 아이집. So 어린이 is a little bit more formal and general. And I think 아이 is more often used to describe some abstract ideas like 아이같은 너 (you are child like) and 어린이 is more tangible, like an actual child. But 아이 and 어린이 are very similar. Hope it helps :)