Is it permissible to give zakat to a child with special needs whose parents are divorced, one of whom is financially well-off but does not provide enough for the child? And how is zakat calculated for stocks with fluctuating values?

The basic principle is that a child is considered wealthy through the wealth of their father. However, if the father refuses to spend on his poor child, then it is permissible for the child's relatives to give him enough from the Zakat (definition). His share of Zakat should be handed over to his guardian. As for Zakat on shares, their value is determined on the day Zakat becomes due and is paid accordingly.