As per the Patron Saint Index, the following are the patron saints of those who have died, including children:
Abel
Barbara
Benedict of Nursia
Catherine of Alexandria
James the Lesser
John of God
Joseph
Margaret of Antioch
Michael the Archangel
Nicholas of Tolentino
Sebastian
As per the Patron Saint Index, William of York is a saint but not currently a patron saint.
There is no patron saint of bullying or against bullying.
Yes he is the patron saint of repented thieves.
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.
Maximus of Rome, who died about the year 250 as a martyr, is not a patron saint as per the Patron Saint Index.
There are patron saints of children or orphans or social workers but no patron saint of child custody.
The patron saint of running water is St. John Nepomucene