1. If you kill off the last member of a house, it can't rebel, for a smaller house you can plan accordingly so you can please other houses more easily, you can also get away with killing the last two of them in a row.
For the same reason you still have to please the bigger houses, you want to scout at least one member of each bigger house with good trails, +15/-10 is the best especially when the -ve match the house's expertise but it is rare, -5 total is acceptable, if all members have -10 total or worst you might want to kill them off unless one of them helps you weak tenet.
2. You will eventually run out of major sinner, to avoid unjust sacrifice, you can purify healthy members with -20 trails, their sin might get worst and now you have new scapegoats, don't overuse it because you might end up with too many Awakened, but more often than not you don't have cage to spare in first place.
3. A sick member always count as unjust sacrifice, but their virtue don't impact tenet, so it can worth it for those +15/-30 sinners, also you save a cage as you often have not enough cages.
4. When you have not enough cages, you can use inquisition on virtue to find out what that sick member up to, also you have to find out the virtue before you can see someone is sick, deal with sick members ASAP as they will make the whole house sick if left alone. On other hand if you have free cage it can act as random inquisition for free too.