Dear Cristinel Stoica,
the content and the writing style of your essay are very appealing to me. You present the problems in a very understandable language, show pros and cons and come to a conclusion. You take into account many of the relevant fields of investigation and many aspects of reality. And you have a well balanced judgement about different / possible answers to the questions posed in your essay. I think you try to tackle these problems as objectively as possible, what especially means to not forget the subjectivity of the agents who undertakes these considerations. I will mark your essay therefore with a very high score.
Allow me to make some annotations to your essay. The question "why is there something rather than nothing?" can be answered in an alternative way. You answered it by a necessity-argument: Maths is necessarily as it is - eternally. I would add a possibility-argument: there cannot be 'nothing' in the radical sense, because this nothing obviously has the potential to produce something. So 'nothing' is a misnomer, it has at least one feature and therefore it can't really exist as something that has no properties.
If we find the complete set of physical laws, these laws would not tell us how they came about or have been selected. They would just build up a consistent overall explanatory scheme where these laws can be thought of as being necessary *from the perspective within the system*, because these laws would heavily relate to each other. Even if mathematics would be the bottom layer of ultimate reality, i doubt that mathematics would be able to explain why this set of mathematical laws should be reserved to correlate with some physical stuff, or stated more precisely, be that physical stuff. Tegmark does explain it the way that all mathematical structures which have some delicate properties *are* indeed physical stuff. This exludes i think a part of mathematics which hasn't these properties, otherwise all inconsistent or contradictory axiomatic systems do also exist as physical stuff. Or think of all infinite series: Does there exist a kind of Hilbert-hotel physically somewhere (as a universe or some other physical instantiation)? I think the MUH demands a certain kind of filtering and for this filtering i know no mathematical law that should govern it (please see my remark on the implications of Gödels theorems in my own essay). So the question for the MUH to be necessarily true seems to remain somewhat subjective to me, because there doesn't exist a metalaw (other than by human definitions) which could decide what mathematical structures are physical by necessity. Why not include all inconsistent mathematical descriptions and let there be an inconsistent (chaotic) universe (or infinitely many of them)? And if so, how can we know that, at the end of the day, we don't live in exactly such a universe (with some delicately hard to detect inconsistencies)? Or even worse: maybe the measurement problem is an indication that we indeed live in such a universe? Maybe the multitude of different interpretations of QM is a hint that we live in a system which is inconsistent at some point and therefore can prove everything, especially all the different kinds of QM interpretations to be somewhat all equally sound and all seem to be equally rational? Surely, if one takes the MUH serious and takes the quest for ultimate reality serious, one should demand consistency as a main property. But does reality necessarily follow this demand? I would strongly answer with yes, but there are enough people out there who would say no, reality is absurd and irrational. Similar to assume ultimate reality to be strictly deterministic and therefore at odds with human experience, these people would argue that existence per se is irrational - and therefore could well be considered as inconsistent (inconsistency therefore would be just another word for them to say that existence is irrational).
So, the MUH demands a kind of filtering of mathematical structures and i see no metalaw which could dictate this filtering. A natural metalaw should be 'consistency', but *how natural* is this in the light of the existence and the possibility of the existence of 'inconsistencies'? Is this demand of consistency only a fluke due to the accident that our universe *seems* to be overall consistent? I would again answer no, but how to prove it? I think the only way out here is to assume that mathematics - and logic - itself are not necessarily eternal facts, but brilliant *ideas* of a higher state of consciousness i would call God. We only think that these ideas are eternal (and in a certain sense they indeed must be - because God itself is eternal and operates beyond space and time), but they could turn out to be only some tiny aspects of such an eternal God. Surely, whether maths and logics are 'temporal' ideas or eternal aspects of such a God depends on how eternity is structured and whether there is a kind of other-dimensional time within it or not. Who knows.