UPDATED:
Legendaries can drop at any time, anywhere, albeit rarely. With the Loot 2.0 updates, new heroes from lvl 1-59 will get a guaranteed legendary drop for the first kill of The Butcher, Belial, Azmodan and Diablo.
Characters below level 11 are unlikely to find Legendary items, because they have a minimum level requirement. Currently there are no Legendary items available below level 11. (source)
For level 60 characters, you should get a guaranteed legendary drop from Diablo every time you kill him after resetting all of your quests and working your way back to him.
Higher tiers of Torment supposedly include an increase in legendary drop rates, but the exact amount of the bonus has not been disclosed. It doesn't seem to be major, though, so you should pick a lower tier of Torment over a higher one if you will kill significantly faster in the lower tier.
Note, however, that some legendaries will only drop in Torment I - VI.
Original answer:
Yes, there are set and legendary items.
Many of them will be available through crafting, requiring the recipe to drop.
Others will drop randomly, but extremely rarely. This can happen at any level, as not all legendaries are high level gear.
The best legendaries will be Inferno drops.
However, it has been stated that the very best items in the game will not be the set items or legendaries. Instead, the very best will be Inferno level randomly-generated rares.
Currently, the answer appears to be no - if such a thing is possible, it's so extraordinarily rare that thus far nobody has encountered it and lived to tell the tale. However, it is possible to craft legendaries using plans found in Inferno - an expensive proposition, but as close to a vendored legendary as you'll likely be able to find.
Best Answer
In patch 1.05 the drop rates were doubled and Monster Power added a new bonus.
Using the base numbers from the Blizzard forums:
It is now trivial to get 100 MF with paragon levels and NV stacks, so I usually assume the 100MF line as the new baseline. With the 300% cap in place, bonuses from NV and bonuses from MP the new max MF is 475% in Normal, Nightmare and Hell (produced experimentally by hitting the cap in Act 1 Normal Monster Power 10). The shrine does not bump this to 500%. I am uncertain if the upper limit is 625% or 650%, as I cannot get 5 NV stacks in Monster Power 10, Inferno, but if it behaves like Normal then the max is 625%, calculated as max. 300% from equipment + paragon levels + 250% from MP 10 in Inferno + 75% from 5 NV stacks.
Empirically, running between 325 and 375 I've been seeing a legendary or set item about every 4 or 5 hours of game play, using Act 3, Monster Power 3. Most of them stink, but that's another discussion all together. I've also had sessions where I've popped 3 in a matter of an hour at MF 260% MP 3 Inferno. I also see a legendary about every full act run at max MF in Normal (sometimes none but usually 1 maybe 2).
Keep in mind that each monster kill is an independent event. The odds are not cumulative, so killing 50,000 white mobs at 475% MF does not guarantee you a legendary pop (though I'd be somewhat startled if you didn't get one). You still only have a small chance on each individual kill to see a legendary or set pop. People often misinterpret the probability of independent events to think that doing more of the same will increase their odds. The odds on each kills stays the same. What changes is the odds against seeing another consecutive failure. See this question and answer for a mathematical treatment.
So let's look at the odds of seeing a legendary or set pop after repeated mob kills.
So what is this graph saying? Looking at the 100% MF for normal mobs line, it says that after about 3,000 white mob kills there is a roughly 20% chance that you will have seen a legendary pop. On the Elite 625% MF line it says that there is roughly an 80% chance that you will have seen a legendary pop after 500 elite kills. It is very important to note that none of these values ever equal 100%, even at 32000 kills the number is still "very close" to 100% but not quite.
Tracking these curves it is very clear that even for large numbers of kills the probability of seeing a legendary pop can be very low for normal mobs regardless of MF, but that it is much better to hunt elites for even less than 1,000 elite kills at any MF. It is for this reason that most serious legendary farmers are getting their 5 NV stacks and killing as many elite packs as they can find, not lingering about with white hordes.
So does this match up with the assertion that you kill 10-30 times the whites is as good as killing elites? Looking at the 60% probability line, at 100% MF against elites, if you are killing 10 times then that's better than the 100% MF against white mobs, it's acutally on par with 300% MF against whites. If you are killing 30 times then the 100% MF normal line is actually above the elite line. My battle net account shows that overall, including levelling at all difficulties I am averaging about 20 whites : 1 elite, just as an arbitrary number for consideration. As a level 60 I target elites, so I suspect this ratio is decreasing but it will take some time for all the levelling to be averaged out.
The maximum pop chance from a white mob at maximum MF is still about half of the base chance for an elite. This fits with my empirical experience. I have popped legendaries off white mobs but far more off concentrated attacks on elite packs.
One thing that changes the scenario somewhat, that I've failed to make note of in this answer is that an elite kill gives multiple pops many (not all) of which have a chance of being legendary. Normal kills only give 1 or 2 chances. As both NV and MP add extra items into kills. If we assume that an elite gives twice as many items as a normal (an easy assumption) then with my white:elite ratio we should only use the 10x comparison, and then clearly the elite killing strategy is modestly better. If we make assumptions about 5 NV stacks or higher MP levels then this gap widens.
It is very clear there are diminishing returns for adding MF. The distance between the lines shows there is a large gain for going from 100 to 300 MF, a smaller but decent gain between 300 and 475 MF but a very minuscule gain for pushing to 625%.
If there are hidden mechanics working behind the scenes or if the base rate is incorrect, then this picture will evolve with time.