Those weren't supposed to be representatives of various races – but specific skill sets. It is the way one puts together a well-balanced adventuring party, after all.
Obviously, but just like TLotR, the intercultural nature is part of the appeal. As is the coherent (in this case sci-fi/futuristic) setting, which this character really doesn't fit into very well. I just really rather would've seen a new and interesting alien instead of a horny reject from an old western.
Larry Niven's Kzinti were well established before this, so they can't count toward another person's obsession with cats.
Of course they can - the show wasn't forced to use them, after all.
Hitting on Kirk a lot might be viewed as, Ha, how do you like it, or provocation to think about role reversal, so not everything in a story has to directly comment on the main theme.
That is an interesting pov that hadn't occurred, and it does make it slightly more amusing in that light, but I still think it's just not a great character. Whether it connects to the main theme or not, it just doesn't add anything worthwhile. And, for the record, one big thing I definitely have noticed in this rewatch is that Kirk's reputation as the insatiable ladies' man really has been exaggerated over the years. This sort of how do you like it concept would've been more fitting with Riker or maybe Paris (or young Kirk in the new movies - but he probably would like it).