These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
The film is definitely critical of the Republican party, but its message is much more nuanced than "Republicans are terrible people." It's a treatise against an extreme nationalist mindset that encourages competition rather than cooperation among the nations (something the Republican party is certainly guilty of promoting, at least in recent years). In the film, an international dick-measuring contest reduces the people of Earth to such mindless, violent, selfish, slobbering monsters that the Nazis end up looking civilized by comparison.
Genius Bonus: Many. For example, all Nazi spaceships have been named after Wagner's operas. The score quotes Wagner's Gotterdammerung in particular.
Fittingly, Gotterdammerung is the German translation of Ragnarok
Adler's name recalls Alfred Adler, an Austrian psychologist who built his theory of individual psychology around the "Will to Power".
When the Indian representative sees the swastika on this incoming spaceships, he quickly hides his own Non-Nazi Swastika.
Moral Event Horizon: When Vivian orders the attack on the Schwarze Sonne, over objections of "What about the women and children down there?". (How the crewknows there are women and children inside a secret Nazi base on the "dark side" of the moon, which they only first saw a minute earlier, we aren't told.)
MST3K Mantra: The characters don't take the space Nazis seriously. Why should you?
No one could ever legitimately believe that Washington is anything but a black man in whiteface. Not much you can do about it, though, besides completely changing his facial features. However, it's arguable that that's a failure at all, since he underwent treatment that made his skin lighter. There's no reason to expect anything other than a black man in whiteface. There are Albino African-descendants, and that is exactly what most people must have confused Washington with.
When the moon mission lander touches down at the beginning of the movie, the engine stops when the foot pads are still a little bit above the ground. (In the real Apollo missions, the astronauts dared not shut off the lunar module's descent engine until the contact light was lit, indicating that the foot pads were on the surface. We lost the Mars polar lander because its descent engine cut out before it was on the ground.)