In wait of a better answer:

You are right in putting quotes around the word "scientifically" here. During the 19th century, scientific racism was a thing, in that one could argue that the reason of inequalities between nations could be caused by genetics and that some races would be inherently more able to live in a technological society than others.

By the 1930s, however, it was hard for scientists to find facts to substantiate that theory, so they had to rely on pseudo-scientists instead, like Alfred Rosenberg or Himmler's mysticism.

The theory of the Aryan race is that the Aryans somehow appeared in India (Himmler was a fan if some parts of Indian culture) which the reality of the Indo-European connection helped substantiate. Their theory was that it was almost "pure" when it reached Nordic countries and then degenerated as it mixed with "lower races".

They did not claim to know how this "master race" appeared. However several theories existed: normal evolution but also alien origin, Atlantis origin. I knew these claims existed but I don't know how fringe they were.

How did they account for Roman domination of the west?

By a mix between "They had nordic ancestries" and "They are not the lowest race so it is normal they succeed when given good conditions". Note that the core trait of Aryans in their theories is not that they are more creative, but more that they are "natural leaders".

How did Hitler justify his black/brown hair and eyes and Austrian background? Was it enough that Austria is a largely Germanic nation?

You may find information about that in this thread Did Hitler hate the way he looked?. tl;dr : the nazis did not pretend that Germans were pure Aryan. They thought their blood was tainted which was the reason for their eugenics politics.