Is it permissible to give charity on behalf of an unknown person who died two years ago by building a mosque, and would that be considered ongoing charity for him? And what are the types of ongoing charity for the deceased?

There is no objection to giving charity with the intention of dedicating the reward to another person, if that person is a Muslim, whether the charity is ongoing (Sadaqah (definition) Jariyah) or not, even if that person has not given charity themselves. Building mosques is considered an ongoing charity. It is better to ask for the reward from Allah Almighty by saying, when giving the charity: "O Allah, reward me for this action and make it a reward for that person." It is also recommended to perform good deeds for oneself.