Original answer, preserved for posterity
Here's the paper.
First, there are some important assumptions that the author - Vyacheslav Dokuchaev - made. Specifically, there are two scenarios:
The particles are charged and the black hole is not rotating. This is not helpful, because all the particles in a given object would have to be charged perfectly. This is not going to happen in an object even as small as a human.
The particles are uncharged and the black hole is rotating. Rotating black holes, described by the Kerr metric, can easily exist, and there's nothing that says that one body can't orbit a Kerr black hole.
So one (or both) of these scenarios must be satisfied.
Second, Dokuchaev mentions life only once outside the introduction, conservatively saying
We hypothesize that civilizations of the third type (according to Kardashev scale [28]) may live safely inside the supermassive BHs in the galactic nuclei being invisible from the outside.
He alludes to life several other times throughout the paper, but barely touches on it. Discovery has really exaggerated the paper's implications: Dokuchaev really has only shown that particles can orbit inside the event horizon in certain conditions. And that's if the paper is entirely correct.
The Discovery article mentions the evaluation of one other scientist, Dr. David Floyd. Floyd says that the paper raises interesting questions, but it also raises some problems;
Astronomer Dr David Floyd from the Australian Astronomical Observatory and the University of Melbourne says even if the theory is correct, it would be impossible to know what is occurring beyond the event horizon of a black hole.
"At this point — and perhaps forever — we're restricted to making untestable assertions," says Floyd.
"As far as we know, matter would go into free fall, that is, it would all fall into this tiny infinitesimal point at the centre which forms the singularity."
Floyd says that one shortcoming of the paper is that it assumes radiation has no impact on orbits inside the black hole.
"It wouldn't take much to produce drag which would slow down the orbits described in Dokuchaev's paper, causing them to collapse onto the singularity".
And that's just talking about particles orbiting inside the black hole - not taking life into account!
I don't think the paper gives a convincing argument at all.
What I'm interested in is how quickly (or slowly) time would pass for this planet taking into account time dilation due to extreme gravity and how much time this leaves for life to develop before this black hole explodes due to Hawking radiation.
We can't use a simple approximation that would be used around Schwarzschild black holes, so we have to go to the Kerr metric. The formula for that is way too complicated to work with.
Another problem with using time dilation inside the black hole is that you end up with a factor of $$\sqrt{-a}$$
where $a$ is less than 1. So we have a bit of a formula breakdown for non-rotating black holes. This may be the same for rotating black holes - impossible to calculate.
For the time it takes for black holes to evaporate, I got $$4.63 \times 10^{104} \text{ seconds}$$ So they'll have a loooong time. Wikipedia gives a much larger figure, though the mass of the black hole is different.
I really don't think that life could live here, though.
New answer
That's the original answer. I want to rework it, because I was in a bit of a rush when I wrote it, and it could be better. So I'll mostly scrap that and start anew. I'll reuse some bits, though.
Correct me if I'm wrong, but here's the setup:
- The setting is a supermassive black hole, like the one at the center of our Milky Way, Sagittarius A*.
- There is some sort of planet or other body orbiting inside the event horizon.
- This body can presumably support life, illuminated by the surrounding radiation.
You want to know:
- The time dilation the planet would experience.
- Whether or not life could develop before the black hole evaporates due to hawking radiation.
- What could happen to jeopardize that life.
Time dilation
For a non-rotating (i.e. Schwarzschild) black hole, the formula for time dilation is
$$t+0=t_f \sqrt{1-\frac{2GM}{rc^2}}=t_f \sqrt{1-\frac{r_0}{r}}$$
The issue for using this for an object within the event horizon is that the term
$$\frac{r_0}{r}>1$$
and so we get an imaginary number.
For a rotating black hole, the equation is more complex, as shown here:
$$\frac{dt}{d \tau}= \frac{1}{\Delta} \left[\left(r^2+a^2+\frac{2Ma^2}{r} \right) e - \frac{2Ma}{r}l\right]$$
where $a = J/M$, $\Delta = r^2-2Mr+a^2$ and $e$ and $l$ are constants. The calculations would take a while and depend quite a lot on the properties of the black hole. If we know $M$ - which we do - and we know the black hole's angular velocity, $\omega$, we should be able to figure it out. That would take a long time, though. Feel free to plug in some numbers and figure out how long you've got.
This and this are also helpful.
Hawking radiation
There are a couple relevant formulas for Hawking radiation: the temperature of the black hole:
$$T=\frac{\hbar c^3}{8 \pi G M k_B} \approx \frac{1.227 \times 10^{23} \text{ K}}{M} \text{K}$$
the temperature of the emitted Hawking radiation:
$$T_H=\frac{\hbar c^3}{8 \pi G M k_B}=T$$
the power emitted by the black hole:
$$P=\frac{\hbar c^6}{15360 \pi G^2 M^2}$$
and the time it will take for the black hole to evaporate:
$$t=\frac{5120 \pi G^2 M_0^2}{\hbar c^4}$$
Doing the calculations shows that the temperature of the Hawking radiation is negligible, as is the power emitted. The time is on the order of $10^{100}$ years, perhaps a few magnitudes higher. That's for the outside world; the people inside will experience less time than that. If you could calculate that. . .
But all is not lost! Black holes - and especially supermassive black holes - generally have accretion disks surrounding them containing hot gas, dust and plasma. As I explained here, the formula for the temperature of the accretion disk is
$$T(R)=\left[\frac{3GM \dot{M}}{8 \pi \sigma R^3} \left(1-\sqrt{\frac{R_{\text{inner}}}{R}} \right) \right]^{\frac{1}{4}}$$
This will give you more energy than the Hawking radiation, and it will prolong the life of the black hole. Supermassive black holes are monsters; they can eat stars and gas clouds. It throws a wrench into your calculations, but it gives you hope for the life.
Things that could destroy life
It also fits into the category of "Any other unavoidable events that will eliminate all possibilities of life." In fact, all of the things in this category have to do with the black hole eating up something or merging with another black hole. The conditions here would be extreme - even more incredible gravitational forces, incredible temperatures, strong magnetic and electric fields. . . The one thing that could destroy this life would be an unfortunate encounter with matter coming into the black hole.
