You know, it is
actually possible to signal from your dreams. For example:
It was Keith Hearne (1978), of the University of Hull, who first exploited the fact that not all the muscles are paralyzed [during sleep]. In REM sleep the eyes move. So perhaps a lucid dreamer could signal by moving the eyes in a predetermined pattern. Just over ten years ago, lucid dreamer Alan Worsley first managed this is in Hearne's laboratory. He decided to move his eyes left and right eight times in succession whenever he became lucid. Using a polygraph, Hearne could watch the eye movements for sign of the special signal. He found it in the midst of REM sleep. So lucid dreams are real dreams and do occur during REM sleep.
Also, for the record:
It is sometimes said that discoveries in science happen when the time is right for them. It was one of those odd things that at just the same time, but unbeknown to Hearne, Stephen LaBerge, at Stanford University in California, was trying the same experiment. He too succeeded, but resistance to the idea was very strong. In 1980, both Science and Nature rejected his first paper on the discovery (LaBerge 1985). It was only later that it became clear what an important step this had been.
But these are just signals. If you would want to actually send messages from your dream, you'd have to first develop some kind of morse-code, or just use plain morse-code, and translate it to eye movements. You'd need to know this code thoroughly by heart. Then, you'd need to monitor your eye movements somehow. The article mentions a polygraph, so you'd have to get one of those that could measure your eye movements.
Suppose you have both of that. Then you would need a signal you send so you could find your message back on the long graph representing your sleep. You could use the eight times left and right for that. Then you could translate the movements back to the original message when you wake up.
But there would be more problems. You can't read text in dreams, so writing it down for easier translation wouldn't work. You'd have to translate it to movements in your head, although I don't know if that is possible in dreams. Also, I don't know the nature of the problems you want the answers to, but I guess they can be pretty lengthy. Perhaps too long to transfer in separate letters.
So, concluding: it would be possible, but only if you're very very determined.
Reference: http://www.psywww.com/asc/ld/blackmor.html, Lucid Dreaming: Awake in Your Sleep?