Is it obligatory to move a child buried in the home's courtyard—due to the inability to afford a grave in the public cemetery—to the public cemetery, knowing that children might pass by the burial site while playing? And what are the implications of that?

There is no harm in exhuming the deceased from the house's courtyard and burying him in the public cemetery, in order to preserve the deceased and safeguard his rights, as the house's courtyard is not a suitable place for that. Furthermore, since necessity compelled this action, there is no blame on the one who does it.