AFAIK the salute is a Roman rip-off. The Nazis made a lot of vague references to Rome in general - I've been told the concept of "Third Reich" placed Berlin as the "third Rome", the first two having been Rome (duh) and Byzantium. They obviously ignored the far older Russian claim of Moscow as the third Rome...

That weird-looking goose-step march seems to have been endemic in Central and Eastern europe, even today. I've read the Wehrmacht specifically banned teaching the silly thing to line units - it was only used in those big parades the Nazis so liked. It's still staple in East European military parades, AFAIK.