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.
The requirement level is defined by the item and every item has an internal level number.
Affix Level Minimum
In Diablo 3 every item has an internal level number, and every affix has an "affix level minimum". If an item's internal level number is below an affix's level minimum, the affix cannot be applied to the item. This prevents the incongruity of attaching extremely powerful affixes to crude, lowly items.
Source: Diablo 3 Signature Series Guide
There is no list with items and their internal level numbers so I guess they are assigned randomly for every item.
But at least legendary items also have a total affix count and bonus affixes. Here is an example:
Blackguard: LVL REQ: 48; TOTOAL AFFIX COUNT 6; BONUS AFFIXES: 2; GUARANTEED AFFIXES: Damage Bunus, Min/Max Damage, Strength/Dexterity, Crowd Control, Reduction
Best Answer
Some legendaries have their own unique effect. If it has that in the crafting window (like the Reaper's Wraps), it will have those special effects. But there's no way to create legendaries that have their own special effects outside of that. You can't even re-roll them with the Mystic.