Speed of light:
Because from our point of view light travels at 300,000,000 m/s.
While it's certainly true that we see light traveling at $c = 3 \times 10^8
~\textrm{ms}^{-1}$, it's equally important to note that it's not just us who see the light traveling at that speed. The speed of light is, in fact, invariant to any observer in the universe, regardless of how fast or slow they were themselves moving through the space.
Please see this link on Length Contraction and Time Dilation to know how scale of your measurements change at relativistic speeds.
Black Hole:
At its event horizon, pulls space towards it faster than the speed of light due to the fact that although you can't travel through space faster than the speed of light, space itself can stretch faster than the speed of light. Then this would explain why we can't observe any light escaping from the black hole.
You're right in the first statement. However, at the Event Horizon, a Black Hole doesn't pull the space (, time and matter) simply to stretch / expand it. It actually warps and shrinks the space (, time and matter) down infinitesimally to what is known as Singularity. The gravitational pull of a Black Hole at the Event Horizon is so strong that anything falling down would need a speed of more than $c$ to escape. And because even light itself can only travel at $c$, it would forever fail to escape the black hole once caught at / inside the Event Horizon.
Light escaping from the black hole.
There are a few things to consider here. So let's ignore light / photon for a brief moment.
Point of view of an observer falling inside a Black Hole:
An observer $O_{bh}$ falling inside a Black Hole would be subject to infinite Time Dilation caused by its immense gravity. Therefore, put simplistically, from the point of view of an external observer $O_{ext}$, it would seem that $O_{bh}$ is nearly stuck at the Event Horizon. However, this is not the same as $O_{bh}$ literally being stuck at the Event Horizon. $O_{bh}$ wouldn't feel anything special happening with their experience of time. They could theoretically still experience hitting the singularity in a finite amount of time.... if they can survive getting spaghettified first, that is.
Inside the Event Horizon:
Now, before getting to anything escaping a Black Hole, we can begin to ask what would happen to light / photon inside a Black Hole. As the following excellent Q&A would point out, anything falling inside a Black Hole must essentially forego its original state and be reduced to Singularity.
Can a sufficiently large black hole be singularity-free?
It's also important to consider that photons have a rest mass of zero. In other words, they can't ever be at rest. However, they interact easily with any other particles with mass, can get absorbed by such particles, and transfer their energy to them, and then that particle with mass can be made to come to a rest.
Hawking Radiation:
Now, the Black Holes are theorised to evaporate in a finite time via mechanism called Hawking Radiation, although neither has this been verified nor has enough time passed for any Black Hole in the universe to evaporate just yet. It can (possibly?) be argued that the Hawking Radiation may be emitted via / as photons, but it's too big a bite for me to chew upon!
However, with these pointers above, here's something that could be surmised on your primary question....
With two mutually contentious subjects:
(a) Infinitesimally small spacetime (Singularity), and
(b) No stillness for photons
It can't be said whether a photon remains a photon once caught in the Event Horizon of a Black Hole (even if it could escape as one later!). With that uncertainty, if we still consider the speculative case of a photon escaping the Event Horizon due to Hawking Radiation (again, being totally clueless about whether it's the same photon that fell into the Black Hole or a new one that was emitted due to radiation), the time that it would / could experience between its capture and escape (which would meaninglessly be zero, anyway) is not so much of a spot of bother as the other issues are.
Finally, as a closing text, you might perhaps like to read more on an unresolved problem in Physics, namely: Information Paradox.
