What is the ruling on intermingling with the children of the paternal uncle's daughter, who is married to the maternal uncle, and is it permissible to marry them in the future? What are the limits of the relationship between siblings and the children of the paternal uncle's daughter and the maternal uncle?

Cousins on the father's side and maternal aunts' daughters are considered non-mahram unless there is proof of suckling that establishes a shar'i (religiously legal) relationship. The women forbidden by kinship are seven, as stated in Surat An-Nisa (chapter 4, verse 23), and the paternal uncle's daughter is not among them. Your paternal cousins, whether adults or close to adulthood, are treated as non-mahram. Therefore, private seclusion with them is forbidden, and complete hijab must be worn in their presence. Marriage to them is permissible. Their calling your mother "grandmother" does not affect the religious ruling. Similarly, your brothers are non-mahram to their paternal cousins if there is no suckling relationship, so private seclusion between them is forbidden, and hijab must be worn in the presence of those who are adult or adolescent among them. Marriage between the two parties is permissible.