They were a regime of human-supremacists who subjugated, killed and ate other sapient species. So yeah, 'cannibal space nazis' is pretty accurate.
Who knows what the intent was! It was a terrible creative decision regardless.
Yes. And for that to work you need to have coherent theme and decent grasp of psychology. It seemed like they were going for pretty interesting exploration of the dark places 'ends justify means' and 'hard men making hard choices' leads to with Lorca... but then it turns out that instead of a psychologically scarred war veteran he was a cartoon villain from mirror universe. And that's the point, that is how Discovery sucked at dealing with the dark themes, it made a joke out of them.
*shrug* I think 'puerile' was perfectly fair. Edgy cannibal space nazis are the sort of villains twelve-year-olds find cool.