Thanks for the interest in the Orphanage Project, Execution. That's quite a list of questions! To be honest, some of them never occurred to me before, so I hope I can do them justice. Here goes:
1. Any LTW is fine.
2. Yes, you can certainly get LTR for the orphans. I went with Steel Bladder and Dirt Defiant for mine; my reasoning was that I wanted them to have LTRs that would benefit them once they left home and were no longer under my control. But you could go with different LTRs if you want. The key is that they need to have at least 25,000 happiness points.
3. The consequences for not meeting the requirements would depend on which requirement it is. If the requirements for aging up aren't met, then you can't age the children up and have to wait for the game to age them up. I gave myself a bit of leeway with the requirement that they have level 10 in a skill. The two painters didn't actually finish level 10 until after they aged up to YA, but they did finish the next day. If your orphans reach YA and are lacking one or more key requirements, if it's something they can still complete, I think you should give them a little more time before moving them out. If for some reason they can't complete the requirements, then your orphanage has failed to meet the conditions of its existence and whatever consequences are called for in your scenario would apply. (For example, in my Riches Galore story, if Pippa and Sebastian had failed with one of their orphans, they would have forfeited their parents' estate.)
4. I'm not sure I understand the purpose of changing the aging requirements or having the orphans adopt orphans. If your original Sim/s are the ones in charge of the orphanage, and if the goal is for them to help as many orphans as possible, it seems to me that lengthening the time that the orphans would spend in each life stage would just reduce the number of orphans that can be helped. Same with keeping YA orphans in the house to adopt. If there's something I'm missing here, let me know.
5. I've never heard of a Bad Apple Sim before! It does sound like fun, and would certainly be a challenge. If you think you can meet the requirements with a Bad Apple, go for it.
I hope I've answered your questions. Let me know if there's anything you're still wondering about. And good luck with your orphanage project!