How are metalogic proofs valid? I'm reading some materials on mathematical logic. I wonder how we can "prove" metalogical properties (soundness, completeness, etc.)? As at this point, the proof system has not been verified yet. Isn't this a chicken-and-egg question (we may then have meta-metalogic)? What types of proofs are considered valid at this stage?
Short answer: yes, it is essentially a chicken-and-egg problem, or perhaps a hermeneutical circle or a spiral.  The common interpretation of the incompleteness theorems is that this circularity cannot be avoided. 

You can use normal mathematical reasoning for the metatheory - which is not formalized in the first place - or you can choose some formal theory. In the latter case, you can choose a stronger metatheory, like ZFC, or a weaker one, like PRA. 

In principle, you would then be able to choose a meta-metatheory, a meta-meta-metatheory, etc., but in practice essentially all the interesting issues arrive at the theory/metatheory level, and the higher levels just repeat these issues rather than giving new issues.
Yes, it's all circular. You can have weird cases (maybe not in math), were you are proving metatheoretical stuff in a different logic (many sorted, polivalued, second order, etc.), but your theory is in first order. In math this never happens, as far as i know, but in logic people do these weird combinations. But again in these cases are kind'a cirular because you have to know the theory of your metatheory, so one assumes it.