It is true that a number of psychologists treat intelligent use of mirrors as evidence of self awareness. But I am not convinced. Animals can gather information about their own bodies via various forms of perception, including, of course, vision. Some can also use a mirror - extending the range of their vision - to get information about their own bodies. But I don't see how that implies that they have any concept of self. My guess is that lots of animals do have something that we might reasonably call 'self awareness'. But I don't know of any serious evidence for this.
Is it possible to establish that dogs dream? If not, are there any possible future developments that could?
I think it probably has been reasonably well established. There is a plausible article about this by Susan Daffon at www.pet-tails.com/LPMArticle.asp?ID=234 Sleeping dogs exhibit a lot of behavioural signs of dreaming: they make running motions, lick their lips and so on. They exhibit rapid-eye movement sleep. And some tests have been done that tend to indicate that what goes in their brains when they sleep is pretty similar to what goes on in ours. The best explanation for all this is that they do indeed dream. As Richard says, this doesn't constitute 'proof'. But it does give us reason to believe.