How can one deal with a cousin who expresses admiration through songs and social media, given differences in appearance and religiosity, and concerns about future kinship problems and mental health, especially after an Istikhara prayer whose result was not positive?

Your paternal cousin is a stranger (non-mahram) and must be treated within the bounds of Islamic law. You have done well in your desire to marry someone of good religious commitment and character, in accordance with the Prophetic guidance: "If someone whose religious commitment and character you approve comes to you seeking marriage, then marry him. If you do not do so, there will be tribulation on earth and widespread corruption." A pious man honors his wife if he loves her and does not wrong her if he dislikes her. Do not accept your cousin as a husband as long as he does not maintain his prayers, for whoever neglects prayer is more neglectful of other things, and "prayer prohibits indecency and evil." Advise your cousin with wisdom and good counsel regarding prayer. If he repents and becomes upright, then seek Allah's guidance concerning him if he proposes to you.