Not sure about hospitals since every house had priestesses, more priestesses = more power. But the "school" is amply described in the Drow books, you follow Drizzt during his school years where he learns to be a fighter.

The school works because the matrons wish it. The school was another way to take out competition and prove that they have better people and stuff like that. It was very important for a house o have the best in school. So the school was just another playing field for them. Becoming a teacher for a male was one of the few ways that he could gain prestige, so they wanted to do that. Matrons rule the society with an iron fist so as long as matrons wanted something it would happen. So the schools worked flawlessly with a high death rate.

More to add, the Drow society had to abide by laws. A house could not openly move against another house. They had to do it in secret, although everyone knew what was happening. If some members of a house survived the onslaught the house that attacked would have been terminated by the other houses.

Now if the house doing the attack managed to do it in secrecy it meant they had the blessing of Lloth and unofficially they were celebrated for their cunning.

My point is that they could not just go around killing others, strangely they had rules that had to be respected.