Spoilers
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I think the rule of the curse is "remember the last person to have died from the curse, and shortly after you die as well". There does appear to be a small time delay; none of the characters involved died immediately after remembering someone and it is usually at the end of the scene in which they do. I don't think you need to remember the name of the previous person exactly - for example, it wouldn't have been possible for Oswald to remember Tony's name because he never knew Tony's real name to begin with. It seems to me you just need to remember the person and who they were in some meaningful sense. Physical items seem to help prompt memories, but I'm not sure they strictly essential per se - not clear to me what physical object Katherine had that prompted links to Annie, for example. I think it's probably more the close emotional feeling certain belongings can evoke, the same way a particular passage of music did it for Rupert.
With Annie, I think she was just unlucky - the curse beat her to her suicide very fractionally. In another world where she was a bit quicker, she might have ended the curse there.
It does seem to me John was very unlikely to have been killed by the curse if he hadn't committed suicide, because the odds of him being prompted to remember Katherine, who really had nothing to do with him, were very low. Since he was capable of being publicly identified, he didn't die as a result of the curse himself.
With Katherine, she dies because she remembers Annie as a result of her conversation with John. I don't think this was a trick by John - it is heavily implied he thinks he murdered everyone and that's why he commits suicide (especially in the latest version with the added dialogue). Rather, it's just misfortune in how events played out.
What @ figures out is that he is about to die, because he read the name of the last recorded victim and remembered who they were - he recognised the name, and she would have been a colleague of his judging by Alan Thomas' briefing, so when the memories came rushing back, @ realised his time was up. The reason he said "by God I hope this works" is because he was always a bit of a loner and didn't think there was much reason for anyone to remember him, so he hoped the curse might die with him and be finished.
00-dream is a curious one. It seems to me that the curse can partly exert an influence on people and cause them to have strange dreams or visions - you can see this especially with Martha's death as well. I don't think 00-dream is written by @ in his full mind, so to speak, especially as it so dangerously discloses his name. My view is that it is the curse influencing him, although that's obviously a fair bit of viewer interpretation. @ hoped the curse would end, he'd hardly be leaving behind clues that would prompt people examining his files to meaningfully remember him on purpose.
The code (43092) is actually the real life code for the music roll for Because of You by Lee Sims for player pianos, the lyrics to which are written down in 00-dream. This mirrors partly the piano roll which started this all off by reminding Rupert of Amelia, which is 03746 I'd Climb the Highest Mountain, also by Lee Sims, which @ clearly found during his research. Maybe he also found 43092. Maybe the curse found it for him. If you know @'s name, maybe the curse is finding you.