Is it permissible for a man who is married to three women and has 15 children to marry a girl the age of his daughters or younger than them, and what is the ruling of the Sharia on that, and what are the obligatory rights of his three wives and daughters?

There is no objection to a man marrying a woman who is the age of his daughters or younger than them. Indeed, the Prophet, peace and blessings be upon him, married Aisha when she was younger than some of his daughters. This is permissible provided he is just between his wives and gives each her due, while observing the right of the new wife to seven nights with him if she is a virgin, and three nights if she is a divorcee, upon consummation. This marriage does not affect his duties towards his daughters regarding their maintenance and custody.