Could the universe be accurately simulated with an infinitely powerful computer? This would mean that every event happens because of what has hapened before it and there is no randomness factor. At a microscopic level, the motion of atoms is a result of the motion of other atoms around it. When an atom or molecules moves, it may hit another atom or molecule which will acquire part of its force. This means that every motion is related to each other. At a macroscopic level the same rule applies. If I behave a certain way, it is because of the education I have received, my manners, my way of thinking, which I have acquired from my family, friends, and other people surrounding me. In short, everything has a reason to happen and nothing pops out of the blue. For example, me asking that question comes from me and a friend discussing this topic. This happened because I am interested in science and this thought crossed my mind when I was bored.
Is this true, or is there a randomness factor?
If it was true, then how comes our world exists as it is? Since the Big Bang, everything must have been uniform, and there is no reason for there to be irregularities, like planets or suns.
Do quantum mechanics add the randomness factor?
TL;DR: Would it be possible to simulate the Universe with an extremely powerful computer? (One that could simulate every particle at the same time.)
NB: some may think that this question is a duplicate of the question: "Is the Universe deterministic?" In my opinion it is not since that question focuses on quantum mechanics, while this one on all aspects of the universe.
 A: Your question actually contains many questions, which are all related but not so strictly so that it is possible to give a full answer to it.

Is every event in the universe related to each other?

There are various ways to answer this question. Straight forwardly, we have observed that there is a finite speed at which information can propagate in our universe, so that events that are happening right now here and somewhere very far in the universe cannot be affecting each other, in the sense that neither can be the cause of the other.  Note that this does not mean that they cannot have a common cause. Causality is a fundamental concept in physics and science at large.  
At the same time, every event will be influenced by everything that has happened in its (causal) past. Or the other way around: the effects of what is what is happening here and now, will propagate, affecting, even if perhaps in the slightest way, everything far enough in time and close enough in space.

Is this true, or is there a randomness factor?
  Do quantum mechanics add the randomness factor?

While this can be true, we seem to observe a certain amount of intrinsic randomness in the universe. Intrinsic here means that it is not because we are missing information, but because the event actually happens without cause. This is of course in the realm of quantum phenomena, like the where a photon will strike your retina or the decay of single atomic nuclei in the banana you had for breakfast. There seems to be no specific cause for the exact locations or times of such events: they are purely random.
Assuming the universe is best explained by the movement and properties of subatomic particles, then there seem to be a pure source of randomness in the universe, and some events are not determined by their past.

If it was true, then how comes our world exists as it is? Since the Big Bang, everything must have been uniform, and there is no reason for there to be irregularities, like planets or suns.

In fact, there is no a priori reason to believe the Big Band had to be uniform. It might come as a surprise to you, but Physicists were surprised to discover just how astonishingly uniform the universe was at the Big Bang! The natural state of things is to get disordered (2nd Law of Thermodynamics), so avery uniform Big Bang is actually a quite unlikely thing. Scientists have come up with various theories explaining what mechanisms made the universe so regular, the favourite being the theory of inflation.
Finally: 

Would it be possible to simulate the Universe with an extremely powerful computer? 

Yes. It already is. We have theories for the universe which we can use to run the simulations. Actually we do this all the time.
BUT the if accuracy of the simulation is what you are interested in, or predicting the future to the minimal details, then the answer is no. And not just because of quantum mechanics.
Even in a universe with no intrinsic randomness, a simulation can only give you so much information, and will not be able to predict the future to arbitrary details. This is because the equations of motions are extremely sensitive to the initial conditions, and slightly imprecise original conditions will produce incredibly different results. Just try and find a video about double pendulums. This is what mathematicians call Chaos.
Since a computer has only finite memory, it will be able to only store a finite amount of data. For the same reason that $\pi \not= 3.14$, any initial condition we will use for the simulation will be bound to be quite different from reality. 
A: The answer to the title question (Is every event in the universe related to each other?) is clearly a no. Some events can't be related to others due to the fact that light has a finite and unsurpassable speed.
A: The uncertainty principle is often confused with the observer effect. The former says that the certainty in position times the certainty in the momentum is greater than some constant.  We think of momentum and position as two different things, but the underlying physical phenomenon may not be.  Of course, none of this speaks to whether or not quantum mechanics is something we can simulate.  As to whether or not some meta-universe has a computer in it simulating ours, some people suggest that since physics is really fundamentally about information, that seems very computer-like to us and may imply that we live in a simulation.  I would counter that by suggesting that our ideas about information theory are what they are because of the kind of universe we live in.
A: Any finite physical system can be simulated by a universal computer. This includes quantum systems, which could be simulated by a universal quantum computer if we knew how to build one.
Quantum mechanics is deterministic in the sense that the state of the whole of physical reality at one time can be worked out from the state at an earlier time given the appropriate equations of motion. But those equations describe a structure that looks a bit like a collection of parallel universes in some approximations. Those universes are not actually entirely separate as a result of various quantum mechanical effects like interference and entanglement. The reason you can't predict what result you will see in a quantum mechanical experiment is that there is no single fact of the matter about what will happen. Rather, after the experiment there will be multiple versions of the outcome that started out in the same state.
See "The Fabric of Reality" by David Deutsch, Chapters 2,5,6 and 9 for more explanation and "The Beginning of Infinity" by David Deutsch, Chapter 11.
A: By event I assume you mean interaction.  There is certainly a randomness factor if our current theory of quantum mechanics is correct.  The most obvious example is radio-active decay, but most any quantum mechanical interaction will include elements of randomness.  
As for the question of relatedness of events, the answer depends on what you mean by universe.  Our universe is observed to be expanding and parts of it (we assume) are moving away from us faster than the speed of light (not excluded by special relativity because it is due to space expansion and not accelerated motion).  Events in that part of the universe cannot influence events in our neighborhood.  Some people don't consider that part of space to be part of "our " universe and for them the answer is yes because at the very least any event will alter the nature of the gravitational field (or space-time metric in GR language) in its vicinity and that could potentially influence events elsewhere.  If we live in a multi-verse then obviously the answer is no.
