All completely correct, but compared to today's Nazis, Hitler's NSDAP still had some socialist elements. Not that this was much more than propaganda, but in the Third Reich there were indeed housing estates and holiday resorts for workers, state job creation measures such as the massive expansion of motorways (mainly needed for war purposes, of course) and the like.
Of course, the NSDAP was anything but a party concerned with the interests of the population, but even the few "socialist" measures imposed by Hitler's regime have been completely abandoned by today's Nazis.
What I mean is that today's Nazis, especially the US variety, are actually almost worse because they are purely neo-capitalist fascists - in this sense, they can really only be called Nazis (National Socialists) to emphasize their ideological proximity to the inhuman Hitler regime - but there are actually no socialist elements left in their political programs whatsoever, not even a hint (same goes for the AfD, a unfortunately quite prominent neo-nazi party in today's Germany).
Besides all the hate and racism, one thing they still have in common to this day is that they always pretend to be "defending" the interests of the working class or socially disadvantaged groups (if they fit their heinous racial ideology, of course) - but today that's even less true than it was in Hitler's day.