Do revelations in dreams come in a particular phase of sleep?
Whether a particular phase of sleep is needed cannot be said with certainty, and Islam makes no rule of it. What is certain is that a true message from God imprints itself so deeply that it reaches a person in any phase.
Sleep research distinguishes several phases, including dream sleep with rapid eye movements. Dreams from this phase are often the ones people remember best. So it is natural to ask whether God sends His messages at exactly that time.
Whether one has to be in a particular phase cannot be said with certainty. What matters is something else. A message that truly comes from God imprints itself so indelibly that it reaches the conscious mind, whatever phase the sleeper happens to be in. Ordinary dreams often fade soon after waking. A true dream stays.
The beginning of revelation to the Holy Prophet Muhammad (saw) shows this too. As the first hadith above reports, he first saw true dreams that came true as clearly as the break of dawn. The power of such a dream comes from the One who sends it.
True dreams are not reserved for prophets. The second hadith above shows that good dreams remain a way in which God gives glad tidings to ordinary believers as well. Whoever experiences this need not worry about the phase of sleep. They recognise the message by its clarity.