Is it permissible to offer a choice in a vow between fasting and giving money, and which of the two is better? And is the paper that a father took from his workplace considered lawful or unlawful?

Fulfilling a vow (nadhr) is obligatory if it was contingent on a matter occurring and that matter did occur. You have the choice between fasting and giving charity, and giving charity is better. It is permissible for your father to take the paper designated for destruction, for whoever takes possession of something permissible that no one else wants, like game or discarded items, has a greater right to it.