Is it permissible to breastfeed a 17-year-old young man so that he becomes a brother by suckling, with the aim of removing the impermissibility of khalwa (private seclusion) and shaking hands between him and a female relative who lives with him in the same house?

Your cousin (son of your paternal aunt) is a stranger to you. Therefore, it is not permissible to shake his hand, be alone with him, or for you to adorn yourself in front of him. However, if your veiling from him and his entering upon you becomes difficult, then you both may resort to fosterage (رضاع). This can be done by both of you being breastfed by the same woman five full feedings, or by him being breastfed by a woman in a way that makes him a mahram to you. This is a dispensation according to some scholars for those in need, such as the fosterage of an adult for whom his entry upon the woman is indispensable and her veiling from him is difficult, like the case of Salim and the wife of Abu Hudhayfa.