What is the ruling on including my name as the father on the birth certificate of an illegitimate child, and to what extent is it permissible to obtain a foreign passport in exchange for supporting him, and what are my duties towards him?

The child of zina (fornication/adultery) is attributed to his mother. If he is officially attributed to you on paper against your will, there is no sin upon you. However, you must document with witnesses that he is not your legitimate son, so that he does not acquire the rights of legitimate sonship. It is permissible for you to spend on him and sponsor him, and there is no objection to receiving a passport in return, as long as it does not lead to a Sharia-forbidden act. Your relationship with him is like your relationship with any other person, and you are not obligated towards him as a father is obligated towards his son. If you spend on him voluntarily, it is a great good deed.