Is a girl's working after graduating from the Faculty of Engineering considered a sin, knowing that her father spent his money on her education and that the family is financially dependent on the help of relatives?

It is better for a woman to stay in her home, but it is permissible for her to go out to work if the work is lawful and she adheres to the etiquette of the Sharia. However, if the work exposes her to private seclusion with non-mahram men or to unlawful mixing, then it is not permissible. She must obey her father regarding going out for work that does not violate the Sharia, but there is no obedience to him in what violates the Sharia.

If your paternal aunt spent money as a donation, she has no right (to claim it back). But if she spent it intending to get it back, then she has the right to claim it back from the one to whom she gave the money.