Primer on feature classifiers for taxonomy and use?

If you are using the same primers, then this is ideal.

Not exactly. Clustering and taxonomy assignment should be kept separate. Even with closed-ref OTU picking you should still attempt to reassign taxonomy, as closed-ref OTU picking is just aligning to the top hit and does not take any kind of confidence metric into account (e.g., other near-top-hits could have different taxonomic labels!). As @colinbrislawn said:

See @colinbrislawn's excellent answers.

Define "best". Unless if you have a mock community with known composition in your run, you do not know the true taxonomic composition and cannot assess whether one classification is better than another. So you are best any one of the classifiers in q2-feature-classifier — see here. For parameters, just use the default settings unless if you have some particular goals in mind (in which case see the alternative parameter recommendations in Table 2 of the paper you linked to).

The classifier is trained on a specific set of reference sequences, and I imagine you will use the same reference sequences for both human and mouse samples, unless if you have some kind of host-specific reference sequences.

As Colin said:

So same database, same classifier.

Do you mean that you want to cite the classifier that you trained for a separate project? Cite the reference database and cite the classification method that you used, but don't cite the other project.

If you used the same primers, then you have nothing to worry about. See the notes here.