These are great questions. There have been philosophical arguments that suggest that it is impossible to know whether dreams occur while we sleep or are just confabulations we create as or after we awake (call this 'dream skepticism'). These arguments fail once we consider all the evidence and use abductive (best explanation) reasoning. When we wake people during REM, they are likely to report dreams. When we wake them during other phases of sleep, they are unlikely to report (or remember) dreams. When we record neural activity using EEG and now fMRI, we see activity that correlates both with the sorts of experiences reported by the dreamer and with the sorts of activity that correlates with similar waking experiences (fMRI cannot get at all of what you call the "granularity of experiences" but see the link below for an initial attempt). Etc. This body of data could be explained away by a dream skeptic, but that explanation would likely look ad hoc and fail to make predictions as good as the hypothesis that dream experiences are real, in the sense that they occur during sleep and have roughly the content people report (though first-person reports of even waking experience can be inaccurate in lots of ways, and since we cannot report the content of dreams while they occur, the inaccuracies of memory come into play... except that people can make reports of a sort--e.g., intentional eye movements--during lucid dreaming: see http://en.wikipedia.org/wiki/Lucid_dream)
Dream skepticism is one version of the more general problem of other minds--how can we know what other people are experiencing from the first person point of view (or that they are experiencing anything). The sort of 'best explanation' approach described briefly above is, I think, the best way to approach the general problem too. We have lots of reason, including lots of evidence, to think that neural activity correlates with (and is, in some sense, the basis of) conscious mental activity. So, we have lots of reasons to think that creatures with neural activity similar to our own have conscious experiences similar to our own. And this view makes lots of accurate predictions and explains lots of other observations. Etc.
For brain scanning of dreams, see: http://www.wired.com/wiredscience/2013/04/dream-decoder/
For a nice philosophical discussion of dreaming, read Owen Flanagan's Dreaming Souls: http://www.amazon.com/Dreaming-Souls-Evolution-Conscious-Philosophy/dp/0195142357