Is it permissible for me to marry the daughter of my paternal cousin's daughter, considering that my brother and I were nursed by her mother (my paternal cousin's daughter) an unknown number of times?

Suckling prohibits what lineage prohibits. Therefore, a child who suckles from a woman becomes her son and the son of her husband, the owner of the milk. This prohibition applies only to the suckled child, not to his siblings.

Your brother suckling with your paternal aunt's daughter makes him a mahram to her and to her daughter. However, this does not affect your mahram status with either of them.

If you suckled from your paternal cousin with sufficient suckling for prohibition, you would become her brother through suckling and consequently an uncle to her daughter. It makes no difference in terms of mahramiyya whether you suckled directly from the same breast as her or from one of her siblings.

If there is doubt about the sucklings reaching five full sucklings, this does not spread the prohibition according to the preponderant opinion. Therefore, you are permitted to marry the daughter of your paternal cousin's daughter mentioned.

Regarding the engagement gift (shabka), if it is part of the mahr, it is returned to the prospective husband. If it is a gift, the preponderant opinion is that the prospective husband may reclaim it.