Is it permissible for someone unrelated to the deceased to build a mosque or give charity with the intention of dedicating the reward thereof to the deceased? And is supplicating for the deceased better than giving charity on their behalf? Are there any proofs for that?

It is not a condition for the validity of charity on behalf of the deceased that it comes from relatives; rather, it is valid from anyone. Both supplicating for the deceased and giving charity on their behalf are permissible, but supplication is better because the Prophet, peace be upon him, guided to it, saying: "When a person dies, all his deeds come to an end except for three: ongoing charity, beneficial knowledge, or a righteous child who supplicates for him." So, it is better for the living person to perform good deeds for themselves and supplicate for their deceased. Nevertheless, if a person gives charity on behalf of a deceased, or fasts for them, or prays with the intention that the reward be for the deceased, there is no harm in that if it is voluntarily offered.