Most of these are straightforward. My brother's dog = the dog that belongs to one brother. My brothers' dog = the dog that belongs to more than one of my brothers. CHILDREN is the odd one. Sine it is already plural without S, the possessive is CHILDREN'S. You would never use CHILDRENS', since S' is used to create a plural possessive, and CHILDREN is already plural.

Cousins already has an s on the end, you can't add another one.

Normally you use S-apostrophe after a plural noun: ("my brothers' books" or "my cousins' rabbit"). With a singular noun, you use apostrophe-S: ("my brother's books" or "my cousin's rabbit"). But when a plural word (like "children" or "people") doesn't end in S, then you use apostrophe-S: ("our children's books" or "the people's choice").

Polly is ONE person or collective. Polly´s Parrot. Our children are ONE collective. Our children´s parrot. Our cousins are TWO or more PEOPLE or collectves. Our cousins´ rabbit. Also: Jane´s rabbit. (ONE person) The sisters´rabbit (The rabbit of Jane and her sister Francesca). (TWO or more people)