I've been going thru this process for the past 20 months. I can sympathize!! Your sister does not have to change the baby's last name. I chose not to. I'm in almost the exact situation. The father would have to take your sister to court in order to get any kind of custody. That's a separate court from the child support court. If they were never married, he doesn't have much chance of taking the child from her. I worried about all those same issues. Is she doing this thru Maximus? That's the agency that I've been working with.
She definitely needs to take him to court. He needs to be paying his share. The sooner the better. The court system takes months. The question about the supervised visits would have to be answered in court I believe. That's not something the child support judges deal with. They deal strictly with money. Although, the percentage of child support paid to your sister would depend on whether he's involved with the child or not. The judge asks if he's involved/visits the child and if he does not, his amount of support would increase. They try to get both parents involved. Hope this helps you a little. If you have more questions, just email me!