I always liked Omega Glory. I liked any episode where you got to see another Constellation-class starship. I also liked the Capt. Tracey character.
I think all that can be said for the various "parallel Earths" sort of episode--Omega, Piece of the Action, Miri--is simply to regard them as just being a "Star Trek-ism."
I'm not quite sure but I think Omega is the only episode where Spock used long distance/hypnosis/ESP to make the girl open up the communicator. I don't think that particular part of his skill-set ever appeared in any other episode, although I can't be sure of that.
I like Catspaw for the ending. I remember watching it as a kid (60's kid here) and being completely befuddled by the entire story, could not figure out what was going on. But then at the end when the power orb is shattered and we get the reveal of the aliens as those little hermit crabby puppet things--a classic Star Trek "shocker." At least to a kid. To me, at the time, there was a real "horror element" to those sort of plot elements.
Also I don't know about the remastered but on the original verson you can see the puppet strings.