Does time stop inside a black hole? How about at its singularity? I'm a layperson who just read Brief Answers to Big Question and in the book, Hawking proposes that:

A typical black hole is a star so massive that it has collapsed in on itself. It’s so massive that not even light can escape its gravity, which is why it’s almost perfectly black. Its gravitational pull is so powerful, it warps and distorts not only light but also time. To see how, imagine a clock is being sucked into it. As the clock gets closer and closer to the black hole, it begins to get slower and slower. Time itself begins to slow down. Now imagine the clock as it enters the black hole—well, assuming of course that it could withstand the extreme gravitational forces—it would actually stop. It stops not because it is broken, but because inside the black hole time itself doesn’t exist. And that’s exactly what happened at the start of the universe.

I think I understand the notion of time dilation as the clock approaches the black hole but I'm having trouble wrapping my head around the idea that time literally stops inside the black hole. Hawking then uses this idea to propose that there was no time before the Big Bang:

As we travel back in time towards the moment of the Big Bang, the universe gets smaller and smaller and smaller, until it finally comes to a point where the whole universe is a space so small that it is in effect a single infinitesimally small, infinitesimally dense black hole. And just as with modern-day black holes, floating around in space, the laws of nature dictate something quite extraordinary. They tell us that here too time itself must come to a stop. You can’t get to a time before the Big Bang because there was no time before the Big Bang. We have finally found something that doesn’t have a cause, because there was no time for a cause to exist in. For me this means that there is no possibility of a creator, because there is no time for a creator to have existed in.

I've done a little bit of googling on the topic and it seems that time doesn't stop in a black hole. Nothing in high school physics or introductory physics in college prepared me for this idea. 
Does time stop at the center of a black hole? If not, why does Hawking present this idea? How else might can we describe the idea of "there was no time before the Big Bang"?
 A: Hawking is using nontechnical language to talk about things that are hard to describe correctly in nontechnical language. He's also not being particularly careful about making fine distinctions.
You need to distinguish between (1) being inside a black hole's event horizon and (2) hitting the black hole's singularity.
Taking the two quotes together, it's clear that Hawking is talking about singularities. When you hit the singularity of a black hole, time stops for you simply because you're annihilated. This is similar to what happens at the big bang singularity: any observer would have been annihilated by the conditions of the early universe in which the temperature and density diverged to infinity.
Time does not stop for you just because you fall through a black hole's event horizon. For a black hole of typical size, you have something like a few milliseconds of free-fall during which you continue to experience time before you hit the singularity. However, you can't send signals to an outside observer once you cross the horizon, and as an outside observer monitors your signals, your signals appear to become infinitely slowed down as you approach the horizon. So to them, it's very much like seeing your time slow to a stop.
A: Well as you approach the singularity time keeps slowing down correct? So as you keep going closer time keeps slowing and slowing. At the point of the singularity time has slowed so much it is now completely frozen. Not for the observer who is at the singularity (they see time completely normally) but an onlooker who is watching you fall into the black hole would notice you are completely frozen in time. For the observer at the singularity (if they could theoretically see past the event horizon) they would see time passing by infinitely fast. So therefor by the time the observer reaches the singularity the universe would be infinitely old. SOO, this can conclude that nothing can ever reach the singularity because the universe would have ended by then. Another way to think about this is that all matter will reach the singularity at the same time: when the universe ends. So if Hawkings theory of radiation is correct then the black hole would have evaporated before anything gets to the singularity.
A: "Time is stopping" is figure of speech. Actual time of observer, sitting on falling into BH object, will not stop. In fact, every object, beginning from some distance, will fall into BH within limited time period.
"Time is slowing" also for rays of light, and only when their direction is outside of BH. "To slow the time" for light is obtaining red shift. Speed of light is the same, but at the same moment, curvature is high. To move against that curvature is to obtain elongated wavelength.
But, it is true that time does not exist inside BH as the concept. The concept of time is to be able to synchronize each and everyones clock following "synchronize procedure". You cannot synchronize any two clocks laying on the event horizon. But, instead, you can only care about single one clock, associated with falling object. Falling object will not freeze in time, only light emitted by that object will froze, and you will observe that light. For outside observer, object will forever lay on the surface, redshifted and distorted by gravitational lensing. You can also see multiple copies of the same object, because event horizon is surface of infinite light rotation. Light can have few rotations and only after get to you.
Because event horizon is surface of infinite light rotation, we do not have the logically correct synchronize procedure for the surface of BH. But we can "syncronize hardly", it is not guaranteed to work, since event horizon has zero thickness. Slightly off-course emitted light will nor circle the BH, but swept inside. Because curvature flow is almost fully directed inside BH, rather then outside or alongside event horizon.
PS. Inside Kerr BH, where inside shell exist, time could become real number again at some point inside BH. But how to measure it? As you approach the second horizon, you will again expirience strong curvature.
Also, speaking of classical BH. The falling observer could live up until he meets the singularity point. Nothing really interesting will happen for him when he crosses the event horizon, except that he will not see anyting from below, only the light falling inside from the outer layers. Not being able to syncronize his clocks will not being noticable anyhow. Because anyway, he is already at near-speed-of-light velocity and by that point started to emit X-rays.
A: If time stops at a singularity, relative to the outside universe, then the very act of a singularity beginning to form would cause the black hole to evaporate due to Hawking radiation bleeding out over trillions of years at the event horizon where time is progressing in a more or less normal fashion.
Reality hates infinities.
