If a child dies on the scene, can a mother make paramedics take her child to the hospital once they arrive?

Depends. If there is chance of resuscitation then the paramedics will transport, however the death of a child is often treated as a crime scene until proven otherwise. Most state protocols do not let ems transport children if they shows signs of obvious death.
Usually.... ..Unless there are injuries incompatible with life, paramedics will attempt resuscitation and transport.
Yes. I don't see why not. The fact of death on the scene is subjective. Often the type of injury may make many feel like death is the only outcome, but I've seen different cases when people survived the worst ever circumstances.