Is it permissible to designate a sum of money, given as compensation for damages to school property in previous educational stages, to contribute to repairs or provide necessities for another government school that is in poor condition, or to donate it to a government hospital to help the sick and the poor?

You must guarantee what you have damaged of school property and what you have not returned of textbooks that need to be returned. You can pay the value of these items to the school itself, or to another school, or to a public hospital, as the important thing is to return them to public funds. If your sister is trustworthy, you may give it to her to act as your agent in spending it on the school for its repair.