Do Vulcans daydream? They can.
After being bonded together as a result of mating and subsequently separating, T'Pol and Charles Tucker III began to daydream about each other. - http://memory-alpha.wikia.com/wiki/Daydream
Can a Changling? Not sure.
There doesn't appear to be any mention of it throughout the run of DS9. I searched through all the transcripts that mention dreams and Odo is never mentioned to have one or comment on one. At one point Bashir tells him 'sweet dreams,' but I think that is just him using the expression, and not an indication that Odo actually dreams.
The closest thing is the dream-like state is the one he shares with Sisko, Garak, and Dax. In this case they are just reliving one of Odo's memories from an alternate perspective.