For example, nowadays, one might say Собака женщины - большая, but not Собака меня - большая(cf. When should I use genitive personal pronouns versus possessive pronouns?).
Was it ever historically correct to use the genitive case of personal pronouns in this context?
While «собака меня большая» in the meaning of possession is impossible in Russian, besides «моя собака большая» you can say «собака у меня большая», or «у меня собака большая» which shows possession as well. It is the inversion of the well-known construction: У меня (есть) большая собака. But in general, don't expect that languages just copy each other. Each has got its own structure and its own way.
Was it ever historically correct to use the genitive declension of personal pronouns in this context?
Wiktionary claims that third-person possessive pronouns его/её/их stem from genitive form of respective personal pronouns. There's no reference to any reliable source, though. No such claim is made for first- and second-person pronouns.
