First: Is it permissible for a son to perform prayers on behalf of his father, whether the father is alive or deceased?

Second: Is it valid for a son to perform hajj on behalf of his father if the latter died by suicide?

First: It is established in Islamic law that a person cannot act on behalf of another in physical acts of worship, such as prayers. Therefore, it is impermissible for a son to perform prayers on behalf of his father, whether he is alive or deceased.

Second: It is permissible for the inquirer to perform hajj on behalf of his father who died by suicide. He should also supplicate to Allah to forgive his father and have mercy on him, for his father's reckoning rests with Allah Alone. However, the son must have performed his own hajj first before undertaking hajj on behalf of his father. This is based on the narration of Ibn ʿAbbas (may Allah be pleased with them both) who reported that the Prophet (peace and blessings be upon him and his Family) said, "Whoever performs hajj on behalf of his parents, or settles a debt on their behalf, will be resurrected on the Day of Resurrection among the righteous" (recorded by al-Daraqutni in Sunan).

And Allah Almighty knows best.