Almost all modern and even ancient political regimes engage in redistribution of wealth, engage in actions, including war, on behalf of their civic polity. Hitler's ideology is meant to benefit of German people in the same way that the Mandate of Heaven is meant to benefit the Chinese. All political ideologies claim to benefit the people they serve for one reason or another. You're not making a great argument here.
Racism, socialism, and authoritarianism are not incompatible. Just as racism, individualism, and authoritarianism are also not incompatible. Hell, individualism and socialism are also not incompatible. So this doesn't mean anything. What makes Socialism, socialism, and Fascism, fascism, is the context they exist in, the rationale behind the actions they take, and the final outcome they strive towards. Nazi's use of "socialism" as can be gleaned from today's review of literature, primary evidence (such as Hitler's speeches, letters, and actual policy) clearly reveal that "socialism" is whatever the hell they want it to be in order to achieve political power. This does not make their political system related to socialism. Try again.
"Socialism! What does socialism really mean? If people have something to eat and their pleasures, then they have their socialism." - Adolf Hitler
Though I suppose he describes your understanding of Socialism fairly well.
This is false. Ideally, most modern western states view the "public good" to be best served depending on the specific good or service needs to be provided. Realistically, modern policy making is dependent on political preferences of various interest groups.
No. This is a statement that in reality, all political regimes exercise force to achieve a political result. Taxes are taken regardless of how you feel about them. The universal nature of taxation does not mean that the entire world is now a collectivist hellhole, though I suppose I can see where a libertarian might make that argument.
I have yet to see an argument for why Nazis and Socialists are apparently closely related.