Jesus, the events in Pulp Fiction were more detached than this (until you’re given a chance to appreciate them as a whole being at the end). If you just watch what’s going on, you can see how one thing leads to another.
As for what was going on with Mikiya – I don’t know for sure, but I assumed that him going into a fugue state wasn’t particularly abnormal, which is why it took a while for Shiki to catch on. Either that or she didn’t think it was so severe that she’d be forced to act, and it’s made pretty obvious via a bunch of visual clues that she’s an isolated person who doesn’t generally go out of their way for this sort of thing. And you know what – I actually like having to guess at this sort of thing. It’s a much better experience that simply sitting there and having shit crammed down my throat.
Finally, why she started caring: Hrrmmm, maybe we’re going to find out more about this, seeing as there are a bunch of other episodes to come, and an attraction between them is pretty heavily implied. It could also be implied that her spilling the water was reminiscent of his life flowing away/the way his body fell when she touched him, and made her think that if she didn’t intervene, she might as well be responsible for his eventual death.
Think about this stuff on your own. That’s what you’re supposed to do.