There was a "palace" attached to the cathedral, or close by if not actually attached. In France these are also called the “évêché” or “archiévêché”, which literally mean “diocese” and “archdiocese”, but they also mean the building where the bishop or archbishop lives.

For example, in Paris, there was a bishop’s palace somewhere around the original cathedral (Saint-Étienne). It’s not really clear where it was, but it was torn down with the rest of the cathedral when Notre-Dame, was built in the 12th century. A new bishop’s palace was built alongside it, which lasted until 1831. Now the archbishop of Paris lives right across the street from Notre-Dame at 10 rue du Cloître-Notre-Dame.

For the other clergy, they lived in the “presbytère” which is just part of the bishop’s/archbishop’s residence (and today they all live in the same building too). Other churches have their own presbyteries, and the same pattern works for other French cities too. I can give an example from where I used to live, in Nantes - the “évêché” used to be right beside the cathedral (and the street is still called “rue de l’Évêché”). One of the other churches, the Basilique Saint-Nicolas, has a presbytery right beside it (which is pretty memorable because it’s a national historical monument).

The bishop and the priests and various other church officials made up the cathedral “chapter” (or “chapitre” in French). The chapter might include deacons, archdeacons, archpriests, the cantor, among others. These are the “secular clergy”, which sounds kind of strange since today we usually think there is a separation between “secular” and “ecclesiastical”. But strictly speaking, the church distinguishes between secular clergy and “religious” clergy, or members of a monastic order.

Cathedrals often had a monastery attached to them, where the monks lived. Their residence was the cloister or the “cloître” in French. For Notre-Dame, for example, the cloister no longer exists, but it was attached to the north side of the cathedral and that’s why the street that’s there now is called Rue du Cloître.

There were plenty of monasteries in the cities and in rural areas that weren't attached to a cathedral. “Monastery” is sort of a catch-all term for anywhere monks or nuns lived. For some reason we tend to think of a “convent” as a house for nuns today, but a medieval convent could also house monks. As you mentioned, the technical terms are “abbey” (headed by an abbott or an abbess), or priory (headed by a prior or a prioress) which is an offshoot of an abbey. They had their own cloisters and chapter houses, and their own church, if they weren’t already attached to one. (In French these terms are abbaye, abbé/abbesse, prieuré, prieur/prieure, and couvent for the general term.)

A parish is the territory of a single church, and the collection of parishes and parish churches is the diocese. The cathedral is the head church of the entire diocese but it’s also the parish church for the neighbourhood it’s in. The diocese might also have basilicas, which are big or otherwise important churches but not as important as the cathedral. Abbeys/priories in the diocese might be subject to the bishop, but maybe not (they might be subject to another abbey somewhere else, or directly to the Pope). Several dioceses make up an archdiocese, and the archbishop has authority over all the bishops in the archdiocese. Paris is its own archdiocese now, but in the Middle Ages the bishop of Paris was technically a subject of the Archbishop of Sens. So, sticking with Paris, Notre-Dame is both the cathedral for the entire archdiocese and for the diocese of Paris specifically, and it's also the parish church for the people living on the Île-de-la-Cité.

This is the basic situations for England as well, and really anywhere else with a Catholic Church hierarchy.

I hope that clarifies things!