Is it permissible to give one's sister from zakat money to buy medicine for her, intending that as expiation for delaying a make-up fast, given that her father is alive, or must it be given to a poor person in the form of provisions?

The expiation for delaying a missed fast is food, and it is not permissible to give its value in money according to the majority of scholars. Nor is it permissible to give its value in money for buying medicine. One can authorize a close relative (mahram) to buy the food and give it to the needy.

Regarding your brother giving you zakat from his wealth: it must be confirmed that you are eligible to receive zakat. If your father is wealthy and provides for you, then you are not considered eligible to receive zakat.