isn't my passport enough to prove my citizenship and my identity?
Yes, we actually can use one document for two purposes like in the passport case to prove both your identity and citizenship, but you need to provide at least two separate documents.
now, my kid has lived in the US for quite some time now. Is there anything else that I need to do?
You have to provide records of your child's school activities or tax records that show that your child was never assigned a social security number.