Is it obligatory to return the stolen money to the father and donate a similar amount on behalf of the cousin who was falsely accused, out of shyness to apologize to the cousin, and with the intention of seeking Allah's forgiveness?

If you were not an adult at the time of the theft, there is no sin upon you, for the Prophet, peace and blessings be upon him, said: "The pen has been lifted from three: from the madman... and from the child until he reaches puberty." However, you must return what you stole from your father or ask for his forgiveness. As for your cousin, you are not obligated to ask for his forgiveness or give charity on his behalf if you were young at the time, but try to be good to him as much as you can.