Time is based off one's perception of it, and as the Dark Hour is a hidden hour of time, if one can't perceive it, then it doesn't exist. This is noted by Akihiko after rescuing Fuuka that Natsuki won't remember anything ("Don't worry, she's not like us, so she won't remember any of it."), and is also a key plot point in the drama CDs. In regards to the bolded, the Persona universe doesn't really have a "soul" in the sense that you're thinking of. It's more like a presence (sometimes called a wave) that generates an energy source, one's life force and though that one's identity. He used up most of his life force to create the seal, but kept just enough around to hang on to keep his promise. This is why he's noted to be extremely tired several times during the ending. It's not just narration either. Characters even note it if you talk to them during the final days that he appears sick/tired. Rest assured tho, he does die on the rooftop, and he's not literally in two places at once even if his lifeforce is. If it's still confusing to you, think of it as how Igor describes it when you get a game over in The Answer: "As proof of one's life, there exists the danger of death. This proof exists as a flame, painfully burning one's life away."
I actually agree (to a slight degree) about the memory loss being explained slightly poorly. It can be easy to forget some of the details that explain it during the course of the game, especially if the player doesn't bother to check out everything. The game makes it pretty clear that Mitsuru remembering the contradictions in her father's death is what brought about the return of everyone's memories. In the bad ending, where they made no promise to meet up on graduation day and their memories (save Aigis, who appears to be unaffected by Nyx due to her mechanical nature) are wiped clean completely, Mitsuru experiences no contradiction and thus does not remember anything. Aigis herself still has the memories, but she makes no attempt to act on them (and thus cause a paradox) during the bad ending.
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