I don’t think there is a clear answer but I’m inferring that roguishness is something a Drow would pick up on the job. A rogue would definitely train at Melee-Magthere in fighting and strategy but be particularly sneaky as they do so.
(Not particularly relevant to this campaign, but I've always kinda wondered why drow houses didn't have a spymaster as well as a weapon master, house mage etc. It seems a natural fit. Though the real answer is probably 'because Salvatore decided it would have made the cast of Homeland too unwieldy'...)
I don’t think he will be. I thought maybe their father could be a rogue of Braegan D’Earthe. I thought it could be a good way of introducing the mercenaries.
There's no reason they all need to have the same father. I think canonically Drizzt's siblings had three different fathers. Matron mothers are fickle in their affections and are under absolutely zero obligation to be faithful, consorts can come and go, or perhaps a matron mother might see a trait in a male she wants to bring into her family.
If any of the fathers are still around, there might be a bit of quiet rivalry for the Matron's favour, or else if they've been turfed out onto the street after she tired of them, they might see the success of their offspring as a way back into comfortable noble life. Hey, I gave you one useful child, i can do it again!
This is a good point. It might be better to have the house Weapon Master be the patron of the previous matron. Maybe he is Father to the high priestess and elderboy. That lets the elderboy be properly foppish. I’m thinking that while the PCs are in the academy the matron hasn’t been able to simply execute him for lack of active agents. Though you’re right about the succession it sets up paranoia on both sides if he is. Maybe he’s a competent warrior and acts as weapon master but cares more about how things look than how successful they are. This flaw might be uncovered by the PCs when they return to the house after graduation.
There's plenty of ways of doing it. A weapon master who is (for instance) a berzerker or a drunken master - to talk in 5e subclass terms - could be a formidable asset to the house in a stand up fight but useless at training the house troops, designing security measures, organising guard rosters etc. Or maybe he has a weakness for a pretty face and only hires/promotes commoner troops he's attracted to or can convince to sleep with him, to the detriment of troop quality and morale. Or maybe he was a double-edged 'gift' from a more powerful house, and getting rid of him or displacing him would be seen as a mortal insult - and he knows it. It could be an interesting challenge for the PCs to try to get the most out of him - when facing off against a rival house who have a weapon master that's beyond the PCs skills, in addition to their own role in the fight they have to keep this guy focused and on-task long enought to take out his rival.
(can be in any of the schools, not just Arach-Tirilith)
The Scholarship kid. A rank gutter commoner picked up from the scummiest streets by a passing high priestess, possibly because of an omen or because of natural talent, possibly just for expendable sacrifice fodder. Wary and untrusting for good reason. Lacking in social graces, no money, gets looked down on, but knows this is her one big chance and fights like a wildcat when cornered.
The Follower. Not the best or worst at anything, from a middle-low ranking house. Deliberately unobtrusive, hates being in the limelight, consciously puts themselves in the background/middle ground. Will glom onto whoever seems most powerful/highranking/likely to succeed and hope to ride their coattails.
The Godbotherer. Best suited for a Sorcere or Melee-Magthere student, a vocal and outspoken follower of Lloth. Fond of bailing up fellow students and grilling them on theology, or inviting them to pray at awkward times. Tattletale and teacher's pet. Has the somewhat condescending favour of the priestesses, and which makes it ... unsafe ... to lash out at him.
The Vessel. Their family owns a extremely powerful magic item that bestows a very unpleasant curse on its bearer so nobody wants to use it. Nevertheless, it's too powerful to go to waste so the house Matron Mother took the least promising scion of this generation and forced the thing on them. Cannot drop the item, and it cannot be removed, even with remove curse. It's powerful enough to give them a significant head up over their fellow students, but at the cost of a slow putrid fleshrotting disease or something similar. They'd love to get rid of it, but know no way to do so other than dying, and know that even if they did, their usefulness to the family would likely be at an end.
The Dealer. Always somehow seem to have a stash of stimulants, alcohol, exam questions ahead of time, hard-to-find reagents and spell formulae. Nobody knows how they get it all. Everyone owes them a favour. Almost certainly in cahoots with at least one teacher, almost certain to come to a crashing end if the wrong other teacher finds out, absolutely certain to try to sell out their customers rather than go down alone when that happens.
The idiot who gets herself pregnant to a nobody fellow student half way through her first year at Melee-Magthere.