Why is contacting Social Security necessary to receive a new Social Security card?
If you do not contact Social Security to update your name change it could prevent your work wages from being correctly posted to your Social Security record. This could lower the amount of Social Security you receive in the future. It could also cause delays when you file your taxes.
Okay if I want to establish my child's U.S. citizenship, what documents do I need to give to  prove that he's a U.S. citizen?
Only certain documents are accepted as proof of U.S. citizenship like a U.S. birth certificate or U.S. passport.