Regarding matter-antimatter asymmetry We all know that the asymmetry between matter and antimatter is a big puzzle in physics. But I don't know why one expects matter-antimatter symmetry in the first place? As in, is there a fundamental principle which suggests that matter and antimatter should have been produced in the same number at the beginning of the universe? 
 A: We  have the standard model of particle physics, which has been well validated by experiments during the last decades or so, to the point that one can say that it is a convenient encapsulation of all the data. The table of elementary particles which is axiomatic in the theory, has again axiomatically a table with the antiparticles, the theory is symmetric in particle/antiparticle except in a very small percent, where there is CP violation. 

In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation, symmetry) and P-symmetry (parity symmetry). CP-symmetry states that the laws of physics should be the same if a particle is interchanged with its antiparticle (C symmetry) while its spatial coordinates are inverted ("mirror" or P symmetry).

It can be shown that this quantum mechanical framework is the one from which  the macroscopic world which obeys classical mechanics emerges.
Then comes astrophysics and the study of the cosmos where the observation show an overwhelming CP violation, as very little anti particle signals can be seen.
In the cosmological models ,as the Big Bang , the Standard Model is assumed to be the underlying model for the generation of particles, and, except for the small CP violation it is automatic that the number of baryons and antibaryons would be almost the same. Instead we live in a baryonic universe .

is there a fundamental principle which suggests that matter and antimatter should have been produced in the same number at the beginning of the universe? 

Not in the very beginning of the universe, but after the generation of the quark gluon plasma, it  is the   standard model , i.e. the fitted theory to current observations, that is the fundamental principle which is not obeyed, and there is a search to extend the theory to more CP violation than it contains now.
It is an open theoretical problem, how to reconcile the  models for the creation of the universe with the standard model of particle physics.
A: There's no reason for supposing that matter and antimatter should have been produced in equal quantities in the Big Bang. It's simply what is the most "natural" (inverted commas because this isn't well-defined and different people might find it different things more natural).
Compare the related question: what is the total net electric charge of the universe? Is it positive or negative? If you had to guess, what would you say? We know there's negative charge, we know there's positive charge, and a priori we don't have any reason to expect there to be more negative charge than positive charge, or vice versa. Accordingly, the most "natural" answer is zero - the universe is not negatively or positively charged. This is the guess most people would give before looking at the observational data.
The same goes for baryon asymmetry. If you look only at the theory (i.e. no observational evidence) then there's very little a priori reason to favor matter over antimatter. Accordingly the "natural" guess is that there are equal amounts of matter and antimatter.
Nobody will say this reasoning is watertight, and you might not agree with it, but many (most?) people find it reasonable.
A: Well, suppose that you're in the office and you go to the coffee machine. You notice that there is an incredibly tiny puddle of coffee left in the pot. The pot is almost completely empty, but it's not totally empty either. Why is this?
One theory is that it could be a complete coincidence -- the pot was going to have some coffee level or other in it, and that's just what it happened to be. Another theory is that your lazy coworker Steve didn't want to finish the coffee and have to make a new pot, so he left a tiny bit in so the next person would have to do it. The benefit of the second theory is that it explains two weird features of the amount of coffee, while the first doesn't explain anything. But the drawback is that you could waste time thinking about something that turned out to be a total coincidence.
Trying to explain the matter-antimatter imbalance is a lot like this. It's a very small imbalance: there are billions of photons out there per baryon. But it's not zero, either. (If it were zero, you actually would end up with some regions with only baryons just by chance, but there would be much fewer than we observe.) Why is this?
Again, one theory is to just refuse to answer the question: say that the number is what it is, and it doesn't make sense to ask where it comes from. This could definitely turn out to be the right answer! But you don't see scientists talking about this option that much, because there really isn't anything else to say. The people who believe this just decide to work on other things. 
Now suppose you decide you do want to try to explain the quantity of baryons, i.e. explain why it's (1) not zero, and (2) much less than the quantity of photons. You might wonder why the first aspect is always emphasized. The reason is that if you believe the theory of inflation, then any imbalance that existed before inflation gets rapidly diluted away due to the expansion of the universe. Thus, under typical assumptions, you begin after inflation ends with precisely no imbalance, so you need a way to build it up.
A: Logic implies that there must be an error in the standard model of particle physics, or in the Big Bang theory of astrophysics, since the former predicts that the numbers of baryons and antibaryons created at the beginning of the universe must be almost equal, whilst the latter shows that there are almost no antibaryons present in the current state of evolution of the universe; but no accepted explanation exists to explain this discrepancy.
Experimental evidence for the validity of the standard model of particle physics is rather strong.
We must infer, therefore, that some development in the evolution of the universe occurred, which current theories of cosmology expressed in the principal - Big Bang - model do not adequately describe.
The inference is that CP symmetry did exist originally, but that the universe then evolved in an unexpected manner, such that the antibaryons no longer exist.
Why this is considered surprising is itself somewhat puzzling. Given that the nature of antibaryons is to mutually annihilate with any baryon with which they interact, it seems on the face of it to be logical that all antibaryons formed in the early universe would by now - an enormous period of time later - have undergone annihilation.
It is generally believed, by proponents of the Big Bang model, that the universe in its early stages was so enormously dense that it was hundreds of thousands of years old before it was possible for light to pass through it, because its density originally rendered it opaque to electromagnetic radiation. Thus it is difficult to understand why physicists have a hard time believing that antibaryons would still exist after that period, since the great density of particles prior to that point gave every opportunity for baryons and antibaryons, in such intimate proximity to one another, to undergo mutual annihilation.
Logic implies that such a period of great density would facilitate the meeting of baryons and antibaryons, in great numbers, with the result being mutual annihilation. And the consequent release of energy would heat up the early universe to an enormous extent, possibly providing an explanation for it continuing to exhibit opacity for such a lengthy period of time, with the increase in temperature counteracting the natural tendency of the expanding universe to cool as a consequence of the expansion. 
Indeed, much might be explained if the evolution of the early universe was accepted as passing through a phase where, instead of expanding and cooling, it was actually expanding and heating up, in consequence of those annihilations. Inflation itself, a highly debatable topic, might be explainable on the basis of this enormous injection of energy into the system, potentially causing a tremendous - and continuing - acceleration of the rate of expansion.
As a mechanism for explaining certain otherwise perplexing developments in the early history of the universe, the amount of energy released by the discrepancy between the original quantity of antibaryons present and the quantity currently observed in the universe (if we assume the discrepancy to be valid, and seek to calculate how much the missing energy is, and where it went) has inevitably been under-appreciated in most cosmological theories.
