Is it permissible for a father to pay the Zakat from his wealth to his daughter to help her pay lawyer's fees to obtain residency or work permits in her country?

It is not permissible for a father to give his Zakat to his son for the son's obligatory maintenance. However, it is permissible for the son to receive Zakat from his father to settle a non-obligatory debt, or if the father is unable to provide for him. Therefore, if you have a debt, there is no harm in taking Zakat from your father to pay it off. As for taking Zakat to complete residency papers, it is not permissible unless you are compelled to reside in that country and your father is unable to support you. But if he is able to cover those expenses and you are compelled to reside there, then this falls under his obligatory maintenance.