You asked a lot of questions there, many of them not fully formed. I've seen another question of yours asking if dark energy has gravity and asserting Hawking Radiation doesn't exist. Dark energy seems to be your thing. You're tempted to use it as a magic bullet to disprove a lot of cosmological theory without understanding it. Let's talk about Dark Energy for a moment.
The important thing to realize is that "Dark Energy" isn't energy in the same sense that light is energy. We don't know what it is. We probably now more about what it isn't. Dark Energy, right now, is just a thing to make equations work. We don't know if it's a thing which can be moved, stored or transformed. It's a physical property of the universe. In the same way that space and time don't get sucked into a black hole, Dark Energy probably doesn't either.
Do not make the mistake of thinking anything which applies to Dark Energy applies to the unrelated scalar, physical property we call "energy". Dark Energy is just a bad name for the negative mass-equivalence force needed to balance out gravitational observations.
Let me try to answer some of the questions you asked.
I imagine that all mass in the universe turns to black holes, as when mass reaches the temperature of absolute zero, it automatically collapses the atoms, or is that true?
We don't know a lot about what happens inside a black hole, keep that in mind.
That depends on what you mean by "collapse the atom". If you mean fusion of nuceli, sure. That happens in normal stars because of gravity, maybe it happens inside a black hole.
If you mean atoms are no longer a nucleus of protons and neutrons surrounded by electrons, that might happen, too. Neutron stars are stars whose gravity is so high, electrons and protons have been fused to form neutrons. Buuut it's not that simple, and strange things happen. Best to read the article.
As for Absolute Zero, the answer is no. Absolute Zero means everything is in its ground state. It does not mean there is no energy. Being in a ground state does not mean atoms fall apart. For example, a crystal lattice can be at absolute zero.
To use an analogy, if you dump out a bunch of sand into a pile it will form a conical shape. This is analogous to a "ground state" with respect to gravity. It doesn't mean it has no energy, but it can't be changed into a different shape by gravity alone. In contrast, if you build a sand castle with that sand, that is analogous to the "excited state". Give it a shove, and it will revert to its ground state, a pile of sand.
Over time, with all the mass in the universe existing as black holes, and all energy having gravity including dark energy, then the dark energy that fell into the black holes would be captured, and the black holes would get larger and larger.
Yes, there are models of the universe which say everything will wind up in black holes, but then evaporate due to Hawking Radiation.
No, there's nothing to suggest Dark Energy falls into black holes or anything else. This is as nonsensical as to say time falls into black holes.
If dark energy is not a constant, which seems unlikely to me, and if it changes over time, and it's density decreases, and it has no pushing effect on black holes, because they suck it in, then eventually all the energy and mass would be contained within the black holes.
This is, again, coming from the wrong idea that Dark Energy is a thing that can be moved around under the influence of gravity. It (probably) is not.
Dark Energy is not energy. It appears to be a property of the universe. Saying it is not constant is as silly as saying any other physical constant is not constant. The Plank Constant isn't going to get sucked into a black hole.
Eventually all energy and mass would be contained in black holes... except for Hawking Radiation and the accelerating expansion of the universe says it will all eventually escape.
I'm trying to understand why anyone thinks that black holes can evaporate when energy has gravity, based on the stress energy tensor of general relativity, and the escape velocity of a black hole is higher than the speed of light. If a photon could pop out, wouldn't it be sucked straight back in again?
And on to Hawking Radiation. This is based on equations which said black holes should have a non-zero temperature, they should radiate some energy as heat. This power is miniscule, a black hole the mass of our Sun would put out 9 × 10^−29 watts and take 2.098 × 10^67 years to evaporate.
If a black hole is receiving no input, it will eventually evaporate. The smaller the black hole, the faster it will evaporate. In the last second of it's existence a micro black hole will pump out 2 × 10^22 J.
The mechanism of Hawking Radiation relies on quantum effects near the event horizon. In simple-ish terms... space is not empty. Particles and anti-particles are constantly popping into existence and almost immediately annihilating each other. Because it's a particle and anti-particle energy is neither created nor destroyed, conservation of energy is preserved, just as
1 + -1 = 0.
But if this happens at juuuust the right spot on the event horizon, or if quantum tunneling becomes involved, one particle will fall into the black hole and one will escape. To the outside observer, it appears that the black hole just emitted a particle. Conservation of energy says the black hole must have lost energy and thus mass.
The universe is weird, but this process is as accepted as anything else about black holes is.
Stephen Hawking does a good job explaining this in A Brief History Of Time. It's a bit out of date, but still a very good read.
There are two theories really, one says the universe will collapse, causing another big bang, and explaining why the big bang happened in the first place. The other says that the universe keeps expanding until the black holes evaporate into photons which fly around forever, never causing a universe again, but this fails to explain why it ever happened in the first place.
We don't have an answer to how the Universe started. There are plenty of hypotheses. I've even heard one from Roger Penrose which deals with the problem of The Big Freeze.
To sum it up, if the expansion of the universe is accelerating, everything will either wind up in black holes or accelerating away from each other too fast to interact. Given enough time, black holes will eventually evaporate. Given enough time, all matter will decay into energy (this is hypothesized, but nobody has observed proton decay).
Once there is no matter, there is no scale to measure distance against. You can redefine the size of the entire universe to be a single point. All the energy of the universe contained in a singularity. Its expansion becomes the next Big Bang. The gravity waves and disturbances from the previous universe are reflected in the Cosmic Background Radiation and answer questions like why all the matter and anti-matter didn't cancel out.
This is based on a lot of speculation and unproven hypotheses. We just don't know. And it doesn't say how the whole cycle started. Science is ok with that, it'll figure it out eventually.
If this is keeping you up at night, may I suggest a career in Astrophysics?