Yes, Saint William of Rochester is recognized as a patron saint of adopted children, foster children, and parents who have lost children.
There is no patron saint of bullying or against bullying.
No, Saint Nicholas is not the patron saint of thieves. He is the patron saint of children, sailors, and the falsely accused.
There is no patron saint of children's health but the Infant of Prague is the patron saint of good health.
The patron saints of unborn children are Saint Joseph and Saint Gerard Majella.
Saint Nicholas of Myra is the patron saint of children, scholars and penitent thieves as well as a number of other causes. For a complete list of his patronages, click on this link.
As per the Patron Saints Index, the patron saints of unborn children are Saint Gerard Majella and Saint Joseph.
I see no specific saint for healing of children. However, St. Brigid of Ireland is considered as the patron saint of healers.
The patron saint against autism is St. Ubaldus Baldassini.
There are patron saints of children or orphans or social workers but no patron saint of child custody.
yes
The patron saint of running water is St. John Nepomucene