Thought I'd jump in because I am sort of fleshing a bit of this out with my own projects.
With regard to races branching off, that is probably your best bet. Only a few dozen thousand years ago, we shared the landscape with another sentient species - the neanderthals. We shared a long extinct common ancestor, yet we quite happily followed different paths of evolution. The neanderthal had some very stark contrasts from us, in their skeletal structure most noticeably. With that in mind, I find it a whole lot easier to understand where dwarves come from. Elves I find a little tricky, because waters tend to get muddied when you try to biologically explain why elves are so fricking uber all the time. I danced around that one by simply downgrading elves to normal mortals like the rest of us oxygen-thieves, with their own set of medical difficulties that don't seem to plague humanity, and some other characteristics that counterweigh the benefits of their longevity (which I think is a little bit of an untouchable elf trait).
This quite neatly dovetails into your second concept of interbreeding. There have been argument for the theory that humans and neanderthals interbred every now and again, but this will probably be largely unproven until we find fossil evidence of human/neanderthal hybrids. It is indeed possible; tigers and lions have been interbred, even though I think the offspring tends to be sterile/infertile, and donkeys and horses have done the nasty as well, giving us the common mule. I have a sneaking suspicion the human/neanderthal theory have a large support base in scientifically minded people who have a phobia of red-heads or have derisive things to say about the rugby team they didn't make selection for. History will decide.
Differences in physiology is all about evolution finding the most energy efficient way of getting people to reproduce more than they die off. So, have a look at typical physiology. Dwarves are short and stocky, hairier than Robin Williams, and are typically fair-skinned. Those could be very good hallmarks of a biological advantage in being strong diggers in cold environs, with a bit of energy being saved in not producing melanin in the skin.
Elves are harder. Depending on your take, such as fleet-footed-ness, excellent sensory faculties, quietness... well that just tells me of a biological need to get the heck out of danger and hiding until the bad things go away. Like I said, elves are hard.
With global outlook, I take the following tacts: dwarves are about as far-sighted as the next cave-wall, but apply highly developed engineering skills to a pragmatic concept of dealing with issues. Solutions to compelx problems can possibly be described by a dwarf in half a sentence. Elves like to avoid problems like the next predator, but with the ability to watch forests rise and fall throughout their lives, they may have a natural understanding of nature, and the nature of nature, than we like to think we have today. They are born to hide and watch. This may still be their attitude to the world. Watch for movement, and learn. Watch our kingdoms rise and fall like the waves on a beach, and hope if they stay quiet long enough, it will all just go away.
That's just my musings. Have a nice day!