Is the paternal cousin considered a brother to the mother due to mutual suckling between them, and does the questioner become unlawful for him after his proposal to her and their marriage?

If your maternal aunt is your mother's full sister, and your male cousin was nursed by your mother, then he becomes your maternal uncle through suckling (رضاعة) and is considered a mahram (definition) to you. However, if your maternal aunt is your mother's half-sister from your father's side, and your male cousin was nursed by your mother with milk that was not from that father, then he is not considered a mahram to you unless there is another reason for suckling. The suckling that establishes mahramiyyah is five full, satisfying sucklings.