Concerning the normalization - ANCOM is self normalizing (see here) in the sense that the sequencing depths cancel out.
But a good sanity check, I would recommend running ANCOM on both rarefied and unrarefied data. If the above concept is truly working, both rarefied/unrarefied ANCOM results should give the same results (with the exception of a few low abundant taxa).
The false discovery control in general is a bit more broad though. Part of the procedure assumes that at least 2 taxa are changing (I personally think it is 50%, but that's for later discussion), and it looks at all possible pairs log ratios.
By doing so, you avoid may of the issues associated with compositionality (since for a given log ratio, you are only focused on 2 species, random fluctuations in other species won't impact the specific log ratio).
My paper here also highlights some of the false discovery issues associated with proportions.