Is it permissible for someone named Abdur-Ra’oof to be called “Ra’oof,” given his inability to change this matter? And is it legislated for a mother to perform the aqeeqah for her adult son, for whom his parents did not previously perform it, knowing that she named him Ra’oof after a dream in which she saw a number of her relatives named Abd Al-Qadir?

There is no objection to you being named "Raouf" without the definite article "al-", because it is an attribute that can be ascribed to a created being. It is permissible for you to perform the 'aqiqah (definition) for yourself if your father did not perform it for you, based on the saying of some scholars who recommend it for someone who has reached maturity and no 'aqiqah was performed for them, and because the Prophet, peace and blessings be upon him, did so after his prophethood. There is no impediment for your mother to perform the 'aqiqah for you with your permission, though it is not obligatory for her. The assumption that something undesirable will happen to the newborn due to their naming contradicting the visions of relatives is incorrect and is a form of forbidden pessimism.