1. OK, I just took a look at the clip where Spock says "Your uniform, Captain. You should make a very convincing Nazi."
Spock was dressed in an SS uniform and they had just knocked out a Nazi so Kirk could also get his uniform and get out of his civilian clothes. So they could both infiltrate.
Spock says the line totally deadpan. So "in universe" I don't think it's supposed to be Spock deliberately making a joke. I think it's an example of Spock being "totally logical" and just referencing that Kirk will be able to disguise himself convincingly enough as a Nazi so they can infiltrate. I didn't find a clip for the prior part of the episode but Spock's uniform has a helmet, which I recall is to hide his pointy ears, and I believe there was dialogue earlier in the episode about how hard it would be for Spock to disguise himself because of his ears. So "logically" it follows directly from that.
HOWEVER--I do believe on another level it's a screenwriter's joke because we see the offended double take Kirk does when Spock says what he says.
So in universe it's showing us that logical Spock can be totally tone-deaf to the emotional content of what he is saying, even going so far as to not really being aware that he could be saying something totally accurate and logical on the literal level, but on another level, it could be very offensive. So it's also another one of those little "moments" in Star Trek where we glean a little insight into the personalities of the characters. That is Spock as the logically brilliant character but emotionally lacking in insight.
But of course it's also intended by the scriptwriter to be a joke, but the "punchline" of the joke is Kirk's offended double take.
Another aspect to this interchange is the context of the whole episode in relation to the Spock character. Gill's argument for replicating Nazi culture is that it is logical and efficient, yet even emotionless Spock who is depicted as totally clueless on the interpersonal emotional level, logically understands that a totalitarian culture is not really "logical" because it lacks the fundamental value of human life and freedom.
So it shows a contrast between the superficial notion of mechanical logic and efficiency which is sort of the surface shallow portrayal of the Spock character vs. the whole notion of Vulcan philosophy which is much deeper and life-affirming and not simply mindless and heartless "logic."
It may also have been a very subtle dig at the sort of militaristic ethos embodied in Star Fleet itself. Very subtle though. Remember these shows were all being produced during the Vietnam war.
So I think there is a lot of stuff going on there on a lot of different levels. I don't think it was "just a joke."
2. As far as the Southern civil war slave episode, they probably have done it on TNG on the holodeck by now, or in one of the spinoffs. If TOS had lasted a few more years they would have undoubtedly gotten to it at some point. Just like they did a Chicago gangster episode, the Nazi episode, Omega Glory episode, they would inevitably have done the Gone with the Wind/Southern Gothic setting for an episode on the parallel Earth where the South had won the civil war, or maybe they would have arrived on a new planet which based its entire culture on Gone with the Wind, and done a little science fiction morality play based on it.