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.
and, apart from the passport, what other documents can be used as my child's proof of identity?
you can use the adoption decree or a state-issued non-driver's ID card, among other accepted documents.