One of these things is not like the others
Please go to the Question #4 post if you haven't had a chance work on this one yet, since the text below contains the solution.
As I wrote in the question, there are a couple of good solutions to this one.
Questions of the "not like the others" type often can have multiple solutions, although the good ones are designed so that only one of the solutions stands out as the best. In a way, this reminds me of the research done by Douglas Hofstadter on how the ability to think in analogies is central to human cognition.
I will leave the specifics of these similarities to a later post. For now, let's get to the the solution to the problem at hand.
The solution that was posted in the comments section was that Quark doesn't belong because he is the only character who is not half-human. Spock, Torres, and Troi each have one human parent and one parent of a different species. Quark, on the other hand, is all Ferengi.
This was the solution that I had in mind when I designed the puzzle, but an alternate solution which seemed just as valid to me was that everyone on the list except for Quark is a Starfleet Officer.
In either case, Quark is the one who does not belong. He is not like the others.