When doing ANCOM analysis with multiple groups, you can run something like Kruskal-Wallis, which will test to see if all of the groups are equal. If you want to want to do pair-wise comparisons you are right - you'll need to manually verify all pairs. With Gneiss, if you have n groups, you can run n-1 tests (by keeping one category as a reference) - but not pairwise comparisons (see explanation here). I believe this is also the case with ANCOM2 since it is also using linear models underneath the hood.
There is not going to be a clear cut answer here -- because no one knows how to properly answer this question. We recommend a minimum of 10 samples because you cannot fit a good straight line with less than 10 observed samples, leading to the errors observed in previous posts
If you have multiple covariates, than the filtering criteria will likely be higher.
Did you look at the coefficients / pvalues from the regression summary?
Gneiss and ANCOM have very different assumptions / interpretations. Gneiss is more of a exploratory analysis trying to find meaningful clusterings. ANCOM will attempt to find individual driving taxa - assuming that few taxa are actually changing.
So no, we don't always expect these tools to agree.