I just did this. Super easy. You need a copy of every page of your old passport and the first page of the new one. You need a copy of your TM 30 and 90 day report. You also need the transfer visa form filled out.
100% untrue. See what Tod said. Your child will need a visa of you plan to leave Thailand and return. There may also be confusion as to whether your child was born here in which case slightly different rules apply.