Yes. Your father led to your family being increased to 5 members.
is just as grammatical as His coat objected to their door being diminished to seven bunches. ____ Are you really asking if it should be family's? Some grammarians insist on that, but the rule is not universally obey
Not really. Not just because of the clumsy grammar but because it has no real meaning. It needs a verb. Something leading to another state of affairs usually requires a form of action that shows how it led to that condition. It is like saying "My goat led to my expulsion from school". There is no clue there (unless you are deliberately trying to convey an absurd sense of mystery). Howe