Is zakat obligatory on a fixed sum of six thousand dollars that does not grow? Is expiation (kaffarah) due from it for missed days of Ramadan fasting, along with making up those fasts? Is zakat obligatory on the monthly allowance a father gives his daughter, or is the father responsible for paying zakat on her behalf?

Your father is not obligated to pay Zakat on your behalf. Rather, you are obligated to pay Zakat on your wealth if it reaches the nisab (minimum threshold) and a hawl (one lunar year) passes over it, by giving one-quarter of one-tenth (2.5%). You are also obligated to pay expiations and other dues from this money if it exceeds your needs. The money your father gifts to you has no Zakat on it if you spend it within the year. However, if you save it, you must pay Zakat on it when one lunar year passes from your ownership of it. Alternatively, you can pay Zakat on all your possessions when the hawl of the original wealth is due, which is better for you and for the poor.