I'm pretty sure this question has been asked before, Here and Here and a non stack exchange explanation can be found Here and another, perhaps the easiest read of the bunch: Here
Now, I've read (not word for word), but I've read a chunk of those answers and I still find it a little confusing without a clear and concise "yes" or "no" - which would be nice, but none of the answers seem to have that, and honestly, I'm not 100% sure myself. Pinu may be spot on right, but I'm not sure.
Now, your 2nd question, that's actually easy and I think you'll see why.
Scenario 1 - objects fall into a black hole and reach the singularity in a finite amount of time. In this scenario the Hawking radiation still happens just outside the event horizon where particle/anti particle pairs form and one escapes the other falls towards the singularity (in finite time) and by some form of quantum magic, the virtual particle steals mass from the black hole (sorry, I couldn't resist attributing Hawking radiation to magic, even though that's probably not how it works).
Scenario 2 - nothing falls into a black hole cause the Time = 0 event horizon creates an impassible barrier, but in this scenario, you have all this matter in "slow time" falling towards the center of gravity but never passing the theoretical event horizon and the same thing happens with Hawking radiation. The Particle - anti-particle pair occurs where one meets matter that's just outside the event horizon and the other escapes, stealing mass from the matter that's in a tight circle just outside the theoretical event horizon.
So, in both scenarios, it takes finite amount of time for a black hole to evaporate by Hawking radiation.
Now it's (theoretically) possible that it takes an infinite amount of time for a singularity to form. That may be true, but it's not true for the Hawking radiation / evaporation.
Now, I feel I should point out that it may not be true that a singularity takes an infinite amount of time to form See here. But it's an interesting question.
There could also be quantum effects that take over due to the apparent ability of individual particles to jump or tunnel and that could overcome any Time=0 barrier at the event horizon, but I'm just throwing that out there as a possible 3rd possible scenario. I'm not an expert.
Also, what I remember from my grad school mathematics some years ago, an infinite "slow down" is not an impassible barrier if it takes place over an infinitely small distance. "how fast does it take a car traveling at 0 MPH to cross a distance of 0 miles" . . . mathematically, the answer isn't necessarily "forever". But I'm not nearly smart enough to give you a definitive answer here.
and for extra reading (though I can't swear by the accuracy of each answer), I found this to be interesting: can a singularity form in a finite amount of time