Until the mid 13th century, Cardinals were quite literally, just Italian Bishops.
Actually, they still are (well, six of them are, the rest are Italian priests).
The Cardinal Bishops are simply the bishops of the seven bishoprics closest to Rome, while Cardinal Priest are the priests of about a limited number of specific churches in Rome or one of those seven bishoprics (the number of cardinal churches have varied over time).
Since 1962 they don't actually perform any functions in those bishoprics/churches, but until then they had to run their bishopric/church in addition to their duties as members of the College of Cardinals (though in practice they delegated most of it). However, while they all are Bishops/Priests in Italy, not all of them where originally ordained in Italy, and in modern times they usually keep their existing bishoprics in their home country in addition to their new one in Italy (and thus usually accumulate quite a few flight miles over their service in the College of Cardinals).
Also note that until 1059 the Bishop of Rome (a.k.a. the Pope) was elected just like any other bishop, that is by consensus among the local clergy, and until 1139 the local clergy and laity (eg. Christian inhabitants in Rome) had to give their approval, or the cardinal college would have to start over. In game terms, that would be "open elective". Of course, might makes right, and several kings and emperors tries to appoint their own Pope, with various levels of success, but in game terms that is best represented with pressing the claim of an anti-pope.
However, the papacy should probably be able to change succession law to "agnatic elective", but with the bishops in the papal state as electorate, and later start dates should use it...