Legion's Beta is supposed to begin before the end of 2015, meaning that it can't be all that far away.
Some very fascinating news came with Blizzcon, particularly in the way that leveling zones and leveling dungeons will scale to the player's level. A player will still benefit from new abilities (presumably through the Artifact system, meaning many will be passive, but still cool) but they'll be able to group up with players of any level from 100 to 109 as they level.
Still, while there will be no official "first zone" the way that we had Frostfire/Shadowmoon in Draenor, I would guess that they'll still be opening up zones on a gradual basis. The good news is that if they do multiple character wipes like they did in the Warlords beta, you won't be forced to take a fresh character through the exact same quests in order to test out new zones.
I'm certain there will be major bugs in this system early on - never take problems in the beta personally, as you're there to suffer through a broken version of the game so that the one people pay for is (mostly) unbroken.
I wonder if we'll have functional Vengeance Demon Hunters by the time the beta is opened up. I suspect that Demon Hunters will be playable from the get-go, given that they had a pretty polished-looking starting experience quest chain at Blizzcon. But the Blizzcon build forced you to go with the Havoc spec, and Blizzard has indicated that they're still working on the design for how exactly Vengeance would function as a tank spec.
That said, I'm sure the Blizzcon build was already weeks if not months old by the time they unleashed it - it was designed as a preview to get people excited about the game, not a test utilizing massive parallelism (the many players) to sift through the game and discover what bugs they could. So it's possible that Vengeance has come a lot farther since then.
The existing specs that are undergoing radical changes (honestly, Combat-to-Outlaw doesn't seem as big a deal as others, though I can see why "Combat" kind of grated on people as an utterly generic non-descriptor) will, I would bet, be playable, though some mechanics might need some iteration. Still, nothing will see as many changes in beta as the Demon Hunter. If your remember what it was like playing a Monk in the Mists beta (or the DK, though I was not able to do that one) you'll know that new classes get some serious redesigns before they go live.
By the end of the beta, the whole expansion should more or less be testable, but in the early days, I'd really only expect one or two zones and possibly (though it's no guarantee) the Demon Hunter starting experience. Dungeons will come later, probably, and raids will come pretty late in the process (and they do those on a rotating schedule.)
As for a beta start date? I really couldn't say, except that tomorrow's patch is probably the last thing they're doing for the live game until after they've launched the Legion beta, which suggests that however long it takes, that's going to be the next priority.
Personally I'm hoping we get it in November, partially because, if I get in, I'm not going to be able to try out much thanks to the holidays (I can't remember the last time I did anything Winter's Veil-related - not that I'm complaining about having a family to visit, of course. It's a good problem to have.)
I think I figured out that the Warlords beta lasted four and a half-ish months. Granted, Warlords had fewer big systems changes to implement, so the Legion one could wind up taking longer. But even if there's a six-month beta, we could get a May/June release if they meet the 2015 goal. Sign me up! (Oh wait, I'm already signed up.)