I’m not normally one for evil campaigns, but this is the exactly sort of thing I could enthusiastically get behind. Great concept.
A few random thoughts etc.
What school do drow rogues train in? Presumably Melee-Magthere?
Don't have the first few levels be just about the schools, and don't have the schools be sacrosant and immune to normal Menzoberranzan politics. Drow family rivalries go everywhere. If your house has enemies who have scions who are students (in instructors...) atthe schools while the PCs attend, then all sorts of things could happen. Sabotaged spell components, interference with exams or trials, biased assessment or rigged tasks, etc etc etc. Hell, even the High Priestess character might interfere, having one of the Matron Mother's children die or fail out of school can only strengthen her hand within the house.
Have you decided if the Sorcere master is going to be the father of the PCs? Or even if any of the PCs have the same father, or if any of their fathers are still alive? While filial affection among the drow is ... limited, let's just say, if a father sees their child succeeding in a House that's failing, they might try to lure them away - perhaps to some mercenary group, perhaps to be a consort in another House. Or they miight try to ride that child's coattails up the ladder.
I assume the elderboy is very formidable in personal combat to make up for his other failings, or else the matron mother would have gotten rid of him? Or is there simply no alternative at this time? Presumably the martial PC is being groomed to take his place eventually. He of course will be aware of this, and will know what it means for his likely life expectancy.
One of the measures of strength of a house is the number of its nobles, especially its daughters. Drow society is matrilinear, I believe the children of male noble children are pretty much commoners(?). Would the matron mother encourage the High Priestess to take a consort of her own and have children? On one hand, it'd strengthen the house, but on the other hand, it'd strengthen the High Priestess's hand within the house. I assume that the matron mother might still be trying to have additional children herself? And of course having one or more of the House's chief combat or magical assets in the later stages of pregnancy makes a house vulnerable. And then of course there's the PCs. They might be positively encouraged or even ordered to take particular consorts for political reasons, or to bind a commoner with useful skills to the House. Dynastic marriages don't get talked about much in drow politics because the Houses are so walled off from each other and unwilling to cooperate, but it doesn't mean partnering can't be a useful political maneuver.
Have you thought much about the history of the house? The previous matron saw a decline from near the council to mid 20s in ranking, but that implies the matron before her was presumably quite competent. What happened to THAT matron mother, and how did the relative incompetent end up taking over? Maybe the current matron was an instructor at Arach-Tirilith or something, away from the house and unable to intervene? The now-deceased incompetent might not have known all the secrets of the house if her predecessor hadn't seen fit to share them. Similarly, during the house's decline, the compound may have been partially closed down or mothballed due to lack of soldiers to guard it, or commoners to fill it, or funds to run it. What could have been going on in the walled-off pieces when the PCs need to re-open them once the house's fortunes start turning again?
This is a great idea, keep us updated how it goes.