Sleeping is a usual, necessary process for memorization and brain rest. It is commonly known that there are two phases of sleeping: rapid eye movement sleep (REM) with high brain activity and non-rapid with low activity and usually without dreams. When you get tired and fall asleep, you enter directly into second, non-rapid phase, and your internal discourse stops. Obviously, this brain state doesn't differ faster processes.
During REM sleep there are some images, dreams in your head, combinations of ideas, memories and worries, which aren't useful, since brain erases most of it. But there is one interesting technic, lucid dream. Using that you can keep consciousness when falling asleep or get it while sleeping. You can control dreams, do whatever you want, even control memories, if you got high skill.
And what about speed of this REM condition? Despite the rapid eye-movements, this condition also doesn't provide high speed of thinking. Sometimes (e.g. during a nap or while awakening) you can have dreams and feel, hear things round you; and you can observe that your dreams and real world go at the same speed. However, when you awoke, thanks to special chemical mechanisms you forget dreams, it seems dim and far - that makes the illusion of fast feelings.
So, the answers are:
No, it is just another kind of brain work.
You can learn lucid dreaming. Also you may just try remember dreams every morning, e.g. I keep dreams diary for a few years and now can remember about 5 dreams per night, when I started from 1 dream per month.
I'd like to add, from mine and friends experience, there is nothing very interesting or useful about remembering or controlling dreams. It is a real trash for mind, although unusual trash.