Is everything that has happened, is happening and will happen just a reaction to the action of Big Bang? I think the question is vague but interesting.
I was wondering if we trace backwards the cause of any event, and then cause of that cause and so on... where will we end up?
Is there just one trigger of everything that is happening?
Think of dominoes.
 A: 
Is everything that has happened, is happening and will happen just a reaction to the action of Big Bang?

All the energy we have now was determined at the Big Bang, but in that model there is a time, the quark gluon plasma and before where Quantum mechanics reigns. Even if one could calculate the enormous number of classical interactions backwards and reach the quark gluon plasma time, determinism ends. Quantum mechanics postulates impose on  calculations to give only probability distributions for an event at (x,y,z,t), which means the outcome of a specific interaction cannot be reversed. Only its probability of happening.
So even though the present events  depend on the original BB, the path  to them is not reversible.
A: The Big Bang lies in the past light-cone of every event in spacetime, so in one sense the Big Bang is the ultimate cause of everything. But this somewhat begs the question because we do not understand exactly what the Big Bang was, and we probably won’t until we solve the problem of quantum gravity. What we do understand is the state of the universe a very short time after the Big Bang and the laws that have governed the subsequent evolution of the universe.
As other answers have pointed out, this does not imply that the universe is deterministic. The answer to that depends on your chosen interpretation of quantum mechanics. In your domino analogy this is like a domino run which starts in one place but has many branches. We know where the domino run starts, but we don’t know whether the choice of direction at each branch point is pre-determined. Indeed, in some interpretations of quantum mechanics there is no “choice” required, because the dominoes fall in all branches at once.
A: We have a classical intuition of events being time reversible. And everything that exist in the universe is there because of the Big Bang.
There are two main issues:

*

*currently we do believe that QM is the fundamental underlying theory, and this tells us that at the QM level events do have a probability distribution, but the interaction itself is not time reversible (causality). So even if you would try to time reverse all calssical interactions, the underlying QM interactions are not giving you this possibility.


No, QFT is not generically time symmetric.

Is QFT time symmetric, and how is it implemented?


*In SR/GR there is something called relativity of simultaneity. Two different observers might observe events in different order, thus causality between the events could be observer dependent (outside the lightcone). Even if you would try to trace back all the events, this would be from your point of view, and other observers could say otherwise (for certain events). There is no universal reference frame, and thus in your case, there might be no universal path of events backwards.


Spacetime causality holds within the light cone.

Causality under relativity
So the answer to your question is, neither of the two currently accepted theories (QM and SR/GR) gives you the possibility of tracing back all the events to the Big Bang, or we just do not know how to interpret them together to give you an answer.
A: The only physical theory we currently possess that is capable of describing cosmology is general relativity (GR). Although we can to some extent interface quantum mechanics with GR, they are basically incompatible, and we don't know how to reconcile them. Therefore it's natural to answer this question within the framework of classical GR.
The way classical GR expresses the notion of cause and effect is as follows. You start with a Cauchy surface, which is a spacelike surface such that every timelike curve intersects it exactly once. (This is the equivalent of fixing a time t in newtonian mechanics.) Then given the initial conditions on this surface, GR allows us to extrapolate forward or backward in time. The extrapolation can fail if you hit a singularity, or if you have closed, timelike curves ("time machines"). A spacetime is called globally hyperbolic if the extrapolation always works. A globally hyperbolic spacetime is one in which cause and effect hold.
An example of a spacetime in which cause and effect fails is one that contains a timelike singularity. Such a singularity can absorb or emit arbitrary energy and information. Standard big bang and black hole models contain only spacelike singularities, and therefore they are globally hyperbolic and allow a sensible notion of cause and effect.
In GR, a singularity is something that is missing from the spacetime manifold. Therefore the big bang singularity cannot be used as a Cauchy surface or as part of a Cauchy surface.
In cosmological models, it's useful to define a time $t$ which is the time on a clock that has been at rest relative to the Hubble flow ever since the big bang. Then if you pick any $t>0$, it defines a valid Cauchy surface (although most Cauchy surfaces are not of this special type).
Therefore the answer to your question is sort of the opposite of what you imagined. The big bang is uniquely ill suited to stating a set of initial conditions for the universe. Any $t>0$ works fine, but there is no $t=0$, since the big bang singularity isn't even part of the spacetime.
An answer by anna v says:

All the energy we have now was determined at the Big Bang

This is wrong, both for the reasons described above and because GR doesn't have global conservation of energy.
Also:

So even though the present events depend on the original BB, the path to them is not reversible.

The Schrodinger equation has perfect time-reversal symmetry. Other answers have given more competent descriptions of the quantum aspects of this question.
A: We don't know.  That kind of thinking is miles beyond what science could possibly achieve.
However, we can point out that all of our popular theories are deterministic.  Even QM is deterministic when you consider the non-observable wavefunction.  It only becomes non-deterministic in the case of the Copenhagen interpretation, which doesn't need to be invoked here.
However, there are many details.  We don't know what will happen when QM and Relativity collide.  We're still working that out.  Nondeterminism might appear there.  There could be boundary conditions to occur -- edges of the universe which interact with the universe after the big bang.  Current theories say this doesn't happen, but space is really big, and we're just infants on the scene with our cute little scientific method.
Its entirely possible that living beings could actually have some agency in a sense which forces the physical world to be non-deterministic (treating them as boundaries which are not fully defined by the state of the physical world).
However, with all of these, note that the wavefunction is non-observable.  It cannot be measured exactly by any known means.  This is at the heart of the wave/particle duality.  So even if the universe is indeed deterministic, we cannot know the initial state completely, so we must treat it as nondeterministic for all intents and purposes.
A: The interaction of the wave packets associated with two subatomic particles is not predetermined.  It can occur at any point where the waves overlap, or perhaps not at all.
