For network analysis, it's more common to filter features based on prevalence than abundance because the so-called rare bugs can play a key role in your microbial community. Statistically speaking, it makes sense to only include features that are "non-zero" enough in our network analysis. Therefore, I prefer prevalence-based filtering.
You can aggregate your feature table based on the taxonomy or phylogeny, depending on your research question. For example, if you want to integrate data from different studies that used different primers, you will have to aggregate data at the genus level so that they're comparable. Note that aggregating data may miss microbial interactions at a finer scale- if you aggregate data at the genus level, you won't detect microbial interactions at the species or strain level.
Note that network analysis at different taxonomic rank levels can produce vastly different results. You may want to do some exploratory analyses and see how robust the results are.
No. As indicated by its name, it is amplicon sequence variants or exact sequence variants. Many bacteria have multiple copies of 16S rRNA gene, often heterogeneous. Different ASVs may come from the same bacterial species. If you construct your network at the ASV level, you need to be careful when you find highly correlated ASVs that share the same taxonomy or close phylogeny. To circumvent this problem, it may be a good idea to aggregate your ASVs by phylogeny which clusters closely related ASVs together.
There's no consensus on which taxonomic rank is best suited for the network analysis. The network analysis is a useful tool, but the results are best treated as exploratory unless validated by experiments.
Some nice perspectives on network analysis:
From hairballs to hypotheses–biological insights from microbial networks
Use and abuse of correlation analyses in microbial ecology
Open challenges for microbial network construction and analysis