Since the process can take some time, is there a way to claim my child for tax purposes until the process is completed?
Yes, you can contact the Internal Revenue Service and ask them for the form W-7A, Application for Taxpayer Identification Number for Pending U.S. Adoptions.
My kid lived outside the States for quite some time, what can I use as proof that he has never gotten a social security number?
You can use his passport or any record that shows he lived in another country. It could be school records or employment records.