Gender, Independence, and Authorship
Does it happen? Is what these women faculty describe true? For those
  of you who have served on faculty/post-doc selection committees, have
  you seen the evaluations of comparative candidates differ according to
  their gender?


Yes. This does happen. It is often hard to recognize - there's a degree of implicit bias at work all the time that's difficult to detect, because rare are applicants with genuinely comparative applications. It's easy to write off something one way or another - the prestige of the journals in question, the conclusions of the paper, etc. But I have, at times, seen it.


  Latino and Black applicants: There were no Latino or African-American
  faculty on these panels, but the reasons the women faculty gave for
  why women should go the extra mile in establishing independence seem
  to also apply to candidates from underrepresented groups. Have you
  seen evaluations differ according to whether candidates were from such
  groups?


While I encounter this less commonly because of the wide underrepresentation of these (and other) minoritized groups generally, I have seen it happen.


  How else might a candidate demonstrate sufficient independence from
  her PI? Recommendation letters can conceivably convey the quality and
  independence of a candidate, but are there things the candidate can do
  herself?


A couple things come to mind:


Being the PI on your own project. These may be small pilot grants or the like, no one expects you to land an R01 or the like. But it suggests some driving of your own research project.
Collaborations with not your PI. Your PI does not have to be on every paper you write, and it's possible to establish side projects etc. with other researchers. This occasionally has to be done with caution, due the the balancing of commitments, but it is possible.
If your job applications end up with a "Chalk Talk" where you outline your research agenda, nailing that. In my experience, this is where most candidates where I am, regardless of their gender, race or ethnicity, fail to establish their independence. You need to learn to tell the story of you the researcher, and how what you've done is, by and large, a logical arc into what you will do as an independent researcher.