No, 'kids' or 'guys' is ragazzi, the plural of the masculine form. Any time you have a mixed group, or a group of indeterminate gender, Italian defaults to the masculine form. Ragazze is only used for all-female groups.
"The girls' cat" is what people would most likely say, although it is an apostrophetical nightmare to write. I agree that something the "The cat belonging to/owned by the girls" is better. English doesn't really use "of the" too much