Where does quantum mechanics come from? Where does quantum mechanics come from? 
If string theory is proved to be the correct quantum theory of gravity but it failed to explain where quantum mechanics came from can we still consider it a theory of everything? 
Can we possibly discover a theory that explains reality in the sense that nothing is left unexplained (in principle)? 
The origin of quantum mechanics is mysterious for me, I know that quantum states live in a Hilbert space and obey a deterministic law of evolution and we only have an algorithm of calculating probabilities but all of this seems so mysterious. I want to understand the why of quantum mechanics not only how to calculate. 
Do you think that any theory that claims to be TOE have to address these questions?
 A: An "acceptable" theory of everything is quite a matter of taste. 
Since all your experiences are grounded in classical physics, you feel that quantum mechanics is unnatural and seek to "understand" it, probably in terms of your classical notions. For eg: Do you ever question Newton's first law... why should objects have a property called inertia?
Some people have accepted quantum mechanics as the fundamental framework and that our naive classical notions are simply an approximation to quantum mechanics in some regime (day-to-day life). If that was your perspective, then there would be no need for a theory of everything to "explain" quantum mechanics. QM would be an obvious input and the theory would have to predict/explain everything else that you observe.
Roughly speaking, any physical theory has to have something you put in (axioms) and some thing you get out (theorems). At the moment, to me, it seems unlikely that we might be able to find a theory which is unique, compelling, and self-evident, without needing any axioms. It would be wonderful if such a theory existed and if we found it, but I think we're going to take a long while before we get a handle on the question of whether that's possible. But hey, that's just my opinion. The concept is definitely fun to ponder about ;-)
A: "Theory of everything" is not actually a theory of everything. Physical theories reach to the depths of matter, but they leave blank spots behind the front line of knowledge. That is, having studied the bricks in every detail, we could still not understand the buildings made of those bricks. There are such things as chaos and emergent phenomena possible, thanks to complexity of systems.
There are many such gaps known, it is more like they divide scientifically known areas to the island regions. Though we know quantum mechanics, we cannot connect it to the biological level, and then to psychics. There are similar gaps in physics itself: hydrodynamics is not yet understood on the molecular level, atomic nuclei are only roughly understood, and so on. The connection of classical and quantum mechanical phenomena is one of those gaps. It needs some better clarification than we have today, even if the underlying physics is fully described.
So, string theory or any other TOE does not claim to solve those problems at all. Those theories all address only the deepest level of matter, that is particles and their possible internal structure. Even if such theory is found, much work is left to be done for physicists and other scientists.
