my personal headcanon (created for the purpose of fic more so than like... interpreting canon. but i do try to stick to stuff that makes sense in canon) is as follows:
during Terzo's reign as Papa:
Sister Imperator and Papa Nihil, along with much of the Clergy, have esoteric religious goals - they probably want to bring about the end times or whatever.
The rest of the Ministry is closer to something between almshouses and a college - the Clergy loosely oversea this, they say those living there should be attending the Black Mass and so on, but they don’t bother to check, and the people who actually run the place day-to-day (mainly Terzo and some elder Siblings, as well as various Ghouls) don’t care about your actual Satanic practice except in so far as they want to support you with your relationship to it. It’s a home for people who need it – Satanic outreach into the community, with the assumption from the top that those who live there will become true believers and expand the church, but genuine care and concern from the humans and Ghouls who work there.
The Clergy are generally off doing Satan knows what in the Ministry, maybe dark rituals but also probably mostly bureaucracy, and whereas Primo and Secondo were very invested in that side of things, Terzo has little time for that, and prefers to spend his work time (when he’s not touring, anyway) in the residential parts of the Ministry. He performs the Black Mass for those who do attend and he takes on a pastoral role that Primo and Secondo only performed when they had to.
Most Siblings attend classes on theology and ritual, as well as being encouraged to keep up whatever secular education they choose (usually remotely), though nothing is required (because compelling anyone to do anything is rather un-Satanic*). Siblings who are very interested in the esoteric side of things tend to take a lot of classes and ascend ranks until they become members of the Clergy - at that point, being on Sister's good side is very important. Primo and Secondo used to scout out potential new Clergy members, but Terzo doesn't bother, having started to lose his faith in Sister's vision of the institution.
*likewise, the habits that we see the Sisters of Sin wear onstage are just costume. There is no dress code in a Satanic ministry, except for ritual garb worn by Clergy during rituals (Papa wears his paints and full regalia for Black Mass).
once Copia becomes Frater:
I think Copia as Frater would probably not be especially interested in the end times stuff, and I think he'd relax a LOT of Sister's rules (especially around how the Ghouls are expected to behave) - but I do think (given the references to Crowley in 'Call Me Little Sunshine,' for example) that the esoteric interests him more than it did Terzo. I'm not sure how that would manifest - perhaps once his purpose as Papa is taken from him, he seeks a new purpose in trying to help the world through Satanic prayer and ritual. I don't know!
Also, re how many people, buildings, etc: how I imagine it is actually based mainly on my experience of Cambridge colleges - so, for example, my college has nearly 1000 people, which I think is probably the maximum I can imagine the Ministry having. I think canonically there's multiple locations anyway - the Clergy is worldwide, I'm pretty sure. Based on the chapter videos, the primary building looked pretty big, but also not exactly bustling, so I think maybe 600 or so people and not all of them live there full time? I imagine very college-y surroundings: accommodation blocks with individual bedrooms and shared bathrooms, a cafeteria and/or dining hall but also small shared kitchenettes, gardens and probably also (this is not based on college anymore) vegetable patches etc to make feeding all the Clergy+Siblings+Ghouls more economical (not like a full farm tho lmao). I imagine that some parts of the building are really old because the Clergy basically bought an old building from like the 1700s, but then they've gradually added bits - some of the Siblings are excited by the aesthetics of living in the old bit of the Ministry, but when they find out the heating is dodgy and everything's kind of drafty, they realise they'd much rather be in the less pretty new building.
Also, to add to what I said above about learning theology etc - I think that if a Sibling isn't studying, they're probably encouraged to chip in with other tasks like archiving, music-related stuff (not like playing in the band but like helping with arrangements maybe, organising sheet music, that kind of thing), and some end up training to be nurses and doctors in the infirmary, becoming cooks etc. I thiink I remember there being a cleaner in one of the chapter videos who didn't strike me as especially Sibling-y, so I'm ready to believe the Ministry just hires cleaners the way most big institutions would - to be honest, it seems like a risky idea to go the convent/monastery route and leave something as important as hygiene and cleaning up to the Siblings. It seems more in character for Sister especially to just say "fuck it, we'll pay people to do this properly".