Call Social Services first. If they think you have a case, you can get a social worker to fight for you for about $25 (period--not per hour or anything, just a flat $25). It sounds like you already have full-custody, effectively, because he's not taking the child 50% of the time. That will work in your favor if you do take him to court for full custody. You might even be able to get an *increase* in child support. It's unlikely that the court will cut his visitation completely, since you don't seem to have any evidence that he's an *unfit* parent, but if you win the case, you'll be the sole legal custodian of the child, and your ex will only have visitation rights. If you have full legal custody, you should have every legal right to change his name to whatever you want. However, I would recommend talking the name-change over with your ex- first, and considering hyphenating both of your last names, rather than completely changing it to yours.
Check Martindale Hubble's website. Most attorney's today take what is called the Multi-state Bar exam as opposed to the older Bar exam, this exam allows them to be able to go into courts in various states and seek permission to act as counsel within that courts jurisdiction