Is it obligatory to spend from one's private wealth on a husband's children from a second wife, and is it obligatory to be equitable between them and one's own children in giving from this wealth, knowing that their father is well-off and spends equally on everyone?

The financial support of all the husband's children is solely incumbent upon him, and you are not obligated to spend on your children, let alone on his children from the second wife. The issue of equality in giving applies only among your children and does not include his children from the second wife.