Are the premonition’s in dreams?
No, that’s more likely reading dreams.
They seem to communicate with me throughout the following day, highlighting parts that the dream already determined.
What you seem to be saying here is that if you have actual sensation — not some sort of illusory, dreamtime ‘sensation’ — then you are conscious? Then maybe you are. I’m not sure if you could be described as sleeping either, never mind dreaming.
But if your experience of pain turns out to be illusory, in the usual way of dreams, then, well, it’s just a dream, right? 
I have experienced pain in lucid dreams. The memory of that pain still lives with me and feels uncomfortable. I’m not sure how that is different from any other pain I’ve experienced. But lucid dreams aren’t regular dreams. I agree that the pain felt in regular dreams seems less real, less like it impinges on consciousness.
I have had premonitions in dreams. I had a premonition of the Kobe earthquake. Two days before it happened.
It’s possible. In my opinion they are to be interpreted as foresight.
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The question of awareness is interesting. Perhaps this stems from the egocentric nature of our perception of the world. That is, we simply cannot imagine something happening to us differently. In dreams, we perceive some of the brain’s analytical work, but motor reactions are disabled. So, when deciding to run in a dream, we can only imagine it, and then imagine what might happen next. This isolation from real contact with the outside world is the main difference between REM sleep and wakefulness, I believe. Any external or internal signal would adapt this analysis to the real situation, but this adaptation is absent, so the dream unfolds according to a fantasy scenario.