I've watched this episode a few times now, and I think I have most of this convoluted plot clear in my mind (with some major suspension of disbelief) but there are still a few tangles I'd like to unravel.
The one I've been working on and I think I've solved is: how did Eurus get hold of Faith Smith's note? How did she even find out about it? I haven't had time to read through the ten pages of this thread, so apologies if this has already been discussed.
Here are my clues (with thanks to Ariane DeVere):
1. "Please, be aware that one of you is a high-ranking police officer. One of you is a member of the judiciary. One of you sits on the board of a prominent broadcaster."
2. "Anyone ever ‘opt’ to remember?"
"Some people take the drip out, yeah. Some people have the same ... urges."
3. "Culverton gave me Faith’s original note. A mutual friend put us in touch."
And from The Final Problem:
4. "Did you know his brother was a station master? I think he was always jealous."
So what I'm thinking is, CS takes the note from his daughter, and for some perverse reason keeps it as a trophy instead of destroying it. He continues to give monthly top-up confessions to his "friends" and at some point boasts that he has kept the note. One of the board members decides not to take the TD 12 drug and hears this confession. My first thought was the one connected to the prominent broadcaster (Moriarty's brother?) but it could have been any of them, really, even the judge or cop. Whoever it was, they took the information to Moriarty. The consulting criminal is in the business of collecting this sort of information in case it comes in useful. When he meets Eurus and she expresses her wish to meet Sherlock incognito (in order to gather her own information), Moriarty arranges for her to get the note so that she can pretend to be Faith and engage Sherlock in a case. She adds her own clues to the original note to enhance her disguise and to test Sherlock's deductive powers. When he misses the "big one" she knows he hasn't recognised who she really is.
What doesn't quite make sense to me, is why CS would give her the note. Did Moriarty make some sort of deal with him, allowing him to continue with his murder spree? Or was Sherlock dangled in front of CS as a potential victim? CS does say "I have a question for you. Why are you here? It’s like you walked into my den and laid down in front of me." So I'm thinking it's more the former than the latter.