Is it permissible to give charity or seek protection for two people with a single intention, such as a person giving charity on behalf of himself and his mother, or a woman seeking protection for herself and her child? And are these actions valid with the intention for two people?

There is no objection to the person giving charity intending its reward for himself and his mother, or whomever he wishes from his family. It is also permissible to recite supplications (Du'as) and remembrances (Adhkar) with the intention of protecting oneself and whomever one wishes from his family, due to the established supplication of the Prophet, peace and blessings be upon him: "O Allah, I ask You for pardon and well-being in my religion, my worldly affairs, my family, and my wealth."