i wonder, could single family by luck, ruthlessness, and power managed to produce pope for dozens generation (through uncle-nephew line) and so completely dominate Rome, that when they fails to produce nephew, they powerful enough to order some niece-nun raised to Papacy.
or alternatively appoint toddler as Pope and her mother as Pope-regent.
Well, if you make Catholicism a minor religion, then by all means this is possible. As a major religion, this cannot fly - not even in the darkest periods of the institution did this happen for long (IX-X Century), and when the Papal seat got back to have some relevance, soon Popes started being often not Italian, let alone not Roman.
As for OP's question - your idea hinges on the idea of parity between males and females which is a relatively new trend in history. So to get this going, you would need a way to introduce it early with hosts of butterflies or, and this is the easiest way to get this, to make Catholicism a comparatively small (thus more flexible) institution: this allows you to get this done in two ways. The first is to keep the idea of Papal ordainment by acclaim (which could've led to a female being chosen, but still) or you'd have to get female deaconhood and higher going.
Either way, this is extremely hard to get - especially IRL, as St John Paul II made it part of the Magisterium and Benedict XVI made it part of Canon Law, too.
