In 1492, Rodrigo Borja was elected as Pope and became Alexander VI, despite the fact that he had recognized children. As far as I know, the Catholic Church wouldn't let someone who had a family become a priest or even less a Pope. Was this different at the time? Or was the Church so corrupted that nobody really cared? Additionally, if "the rules" didn't say anything about if a Pope could have children or not, could there be a case like his nowadays, or have the rules changed?
apologies for typos and formatting. am on mobile.
short answer: lies & bribery
long answer: for a very long time, borgia pretended that his children (with vanozza dei cattanei) were his nieces and nephews and were fathered by cattanei's husbands. it was so much easier for borgia's backers to believe what they knew were lies because it gave them a bit of plausible deniability when the accusations started. one other daughter that i know of was also claimed to be the daughter of his mistress, guilia farnese's, husband. so leading up to and through the early years of his papacy, borgia never had children as far as most people were concerned. and if you thought differently, you kept your mouth shut because borgia money had a lot of influence.
when innocent viii died, there were only 3 viable candidates for his replacement. the college of cardinals' constitution had been given a serious overhaul to prevent nepotism and royal influence from buying its way into the papacy (oh irony). ascanio sforza, who wasn't very well liked aside from the millanese cardinals, and giuliano della rovere, who was only supported by the french. borgia had been planning his career path for years, if not decades. and pope innocent's death didn't really come as a surprise, so he had time to shore up his alliances. a lot of money, bribes, and promises were floating around the cardinal college. sforza himself was given four ass loads of silver (this is not an exaggeration, he got four mules worth of silver.) to throw his weight behind getting borgia elected. sforza and della rovere never had a real chance to begin with. they weren't particularly keen on the job description beyond the title, and they were just as open to bribery as anyone else. sforza became the vice chancellor of the church for his willingness to play along. and della rovere was given something like 300k ducats from france and genoa just for throwing his name is the ring.
edit: second part of the question
the catholic church is very into their image these days. most everything that comes out of the Vatican is carefully crafted to appeal to the public and tug at their emotions. even the negative press regarding priests and boys. they would never allow a cardinal who had fathered children to even be considered for the papacy. maybe, if a particularly sympathetic priest has taken on an indigent ward or something, but i seriously doubt it.