My response here isn't critiquing anything you've said, rather I'm using the examples you raised, to explore ...
I'm aware of the concept of synchronicities, etc. At the most basic level, the Occam's Razor explanation is that they will always just be "coincidences", or instance of a person attempting to assign meaning to events and find patterns amongst the events. Since they may involve nothing more than a person's mind and reasoning, they are essentially and collectively easily dismissed by many as products of imagination. Thus: natural, but not in an extraordinary sense. This doesn't mean that there isn't something to them, just, at the most basic Occam's Razor explanation, they are typically not extraordinary. There would need to be signifficant enough events as to almost defy coincidence or chance.
Prophetic or precognitive dreams are a bit more interesting, depending on factors. The more detailed they are, the more extraordinary they are, etc ... the more they may show *something* at work between how humans are able to make sense of events in the natural world extraordinary ways. But conversely, if they are riddled with too much symbolism and not enough detail, or are vague, then they are easily passed off as well as being nothing more than the imagination. However, the more extraordinary examples may show an interesting relationship between the way humans perceive information and events that unfold in the environment. Still natural, but if it's a legitimate phenomenon, the mechanism by which it works does not "stand out" apart from a human being as being the conduit, so to speak.
Physical healings are obviously more extraordinary, depending on context. But it's still taking place in the "natural", although the origin of the mechanism at work may point to something else, it's still reflected in the "natural". However if the healing isn't that remarkable (i.e. "My headache got better," or "My cancer got better over time,") ... then the Occam's Razor explanation would be more mundane causes behind the healing. If a limb were to grow back in the space of a minute (or perhaps even in the space of a couple of days) then that is obviously more extraordinary, but still ... it's taking place in the natural. Whatever the origin is, it's reflecting in the natural world.
IMO, if we consider the idea that extraordinary examples of each of those phenomena are legitimate and actually point to a mechanism at work that involves some "behind the scenes" entity or agency which most people would call "supernatural", I still don't see why it's necessary to make that leap. Because even if all those examples WERE legitimate (and not just imagination, or other more mundane explanations), and we simply couldn't explain fully the mechanisms behind them, I still don't see why the jump needs to be made to "it's outside of natural physics !" etc and so forth.
Personally ... and I'm somewhat assuming a lot of things here about you (hopefully I'm not overstepping) ... but I'd bet that if you experienced more extraordinary phenomena that involved events that took place separate from your "mind" so to speak ... like stigmata (effects the body in a dramatic way), or meeting a person whom looked like a human being but was able to do things that defied physics as you knew it (i.e. translocate, or make cars appear and disappear in an instant, vanish, etc), then you *may* see a more direct link to the natural world and things ascribed to the "supernatural". The line between the two may lead one to conclude there is no line. Synchronicity, dreams ... still essentially deal with the mind. Which can lead one to conclude, "I'm either making this all up in my head, or I'm onto something," ... while things physically effecting your body, or happening to the environment around you apart from your mind ... are more extraordinary and may have a different effect on one's conclusions. Just a guess.