As to Naziism's philosophical underpinings... well, there really aren't any. There were philosophers actively associated with them (Heidegger) or besmirched by association (Nietzsche, Husserl) but the Nazis themselves were catch-as-catch-can opportunists aimed at taking power. They got funding and support because they seemed like the lesser of two evils when contrasted with the Commies.

There's a very broad case to be made for a movement of everyone's philosophy away from God towards Nature and Reason during the Enlightenment and then towards moral relativism and hero worship in the Romanticism following Napoleon. Still and all, the Nazis' ideas of the strong as wolves and the weak as sheep is one of the oldest threads in human thought. The classic exposition is Thucydides's Athenian ambassador but wolf cults seem to have been a thing at least as early as the Indo-Europeans.

Mostly when they talked about ideas, though, they relied on junk racialist science and occult nonsense, not proper philosophy.

It's not so much that not that Übermenschen weren't what Hitler had in mind so much as it is that Hitler essentially epitomized the very opposite of what Nietzsche believed and stood for, down to antisemitism. (As a philologist, Nietzsche was very much in favor of the Old Testament as opposed to the New, where he thought it strange that God had found it necessary to learn Greek and then done such a bad job of it.)

His sister and her administration of his estate didn't help his legacy, though.

Rousseau's sensibility as an appeal to the heart, along with his political philosophy. Hitler is a clear outcome of Rousseau.