Borderlands 2 Weapon Leveling

Low. Really really really low. Legendary weapons drop with single digit percent chances at best, and that's from the proper boss (though some bosses seem to usually drop a legendary the first time when fought only).

There's no single percentage though, different items have different methods of obtaining:

Every time a Burner-type enemy drops a sniper rifle, the sniper rifle dropped has a .88% chance to be the Cobra.

Though the Infinity has a higher chance to drop from Doc Mercy, the chance of an actual drop is still staggeringly low, making far rarer to come by then its other Legendary brethren.

Lots of people seem convinced the Infinity has a lower drop rate but I haven't found anywhere that lists drop chances formally (either from the game's code or crowdsourced experiments), and the Cobra's drop rate and method was released publicly by a gearbox employee (because no one had found it...)

There's a rumor going around that the legendary drop rate from a boss is 1/20 but from my own observations this seems optimistically high. Certain high-drop rate bosses like the Warrior seem to drop one about this often, but at best they'll drop one of their legendaries at this rate rather than having a 1/20 chance for each. Which means you'll probably have to fight Warrior >100 times to get one of each legendary he has, if you're lucky.

No. Drops in Borderlands 2 are almost always chance-based. In particular, legendary item drops usually only have about a 1 in 30 chance of actually dropping.

Gearbox Software have actually blogged about the drops recently. See this article, and its follow-up.