Hi @Alex_14262,
Yes, the OLS method in gneiss is essentially a GLM -- we're performing something very similar to multinomial regression, but using the ilr transform instead of the alr transform.
There's a few null hypotheses - a global null hypothesis and local null hypotheses. The global null hypothesis is does the overall fit explain the data? For now, we have a measure of R^2 to handle this, but it would be nice to have a global F-test / pvalue to evaluate the overall fit -- just haven't gotten around to it yet.
The local hypothesis evaluates the likelihood of the coefficient of a balance being zero. If the slope is close to zero, then that balance isn't very explanatory for the particular covariate of interest.
See here on a recent discussion on variable selection.
Those scatter plots were originally designed to give a high level overview of the overall fit with the top two balances. The percentages represent the variance explained as explained in the tutorial @colinbrislawn linked.
The tree is what you passed in during hierarchical clustering -- however phylogenetic trees can be passed in as soon as this issue is resolved