1. You defined Rightist thought as focused on "heritage, hierarchy, and structure". The Chinese and Russians were, indeed, not big fans of their cultural heritage, but they were extremely hierarchical and structured. There are other reasons why I would categorize them as rightists as opposed to leftists (Trotsky was a Leftist, for example, because he believed that states themselves were invalid, unlike Stalin) but quibbling over that is pointless; my point is that the ideologies most "reactionary" thinkers uphold as "traditional" are actually based entirely in Modernity. So moving on to the group you fully yield as being Rightist:Communism and Stalinism and Maoism are not rightists. They actively deny the value of the natural hierarchy. National socialism doesn't; it actively went out of its way to try and instill a love of the national heritage pre national socialism, and glorification of the natural hierarchies. I haven't read much on Falangism so I honestly cannot comment on that.
2. The Nazi Party was not "traditional" by any sense of the word: in fact, they trampled all over the traditional values of the Germanic peoples, beginning with the idea of the people of Germany being one united kind: the Aryan race. The concept of a pan-Germanic identity, instead of the residents of individual kingdoms identifying themselves as members of those regions, was born out of the Prussian desire to become a great power in Europe- a goal they sought to accomplish by uniting the German-speaking states of the Confederation (which at the time was a loose political and trade alliance) into a single entity. Seeking to create a stronger bond than the loose alliance of the Confederation or the HRE without the resentment of a client state, von Bismark emphasized the "common history" of the Germanic peoples on the stage of Europe, elevating pagan myths and using universal suffrage and the formenting of resentment against the Catholic church (the traditional faith of Germany at the time) to forge the German state out of a couple dozen small states. Nazi concepts of racial categories were not traditional in the slightest, neither in how they carved up ethnicities, nor in their idea that the most "inferior" races should be eliminated. Traditional Germanic ideas of ethnicity showed little concern for skin tone and a lot more concern for religious sects and wars, and there was no belief that other ethnicities should be destroyed: conquered and converted, yes- but not genocided.
Nazi policies such as Lebenborn also flew in the face of the "natural hierarchy" of the family you so laud: women being paid to have sex out of wedlock with "Aryan" men and having their children taken and raised by other families, the kidnapping of "racially-pure" children from other countries to be raised in group homes, and mandatory abortions for the disfigured all fly in the face of traditional views of the family, the State's role in the family, and the sanctity of life. Even the Nazi concept of low-brow and avant-garde "degenerate art" as an inherent moral hazard requiring total annihilation traces back to the Jewish critic Max Nordau's work Degeneration: a modernist philosophical tract.
The Nazis draped themselves in the colors of tradition and reaction, but had no real interest in the traditions of Germany, as they were in direct conflict with Nazi goals. Instead, their ideas were rooted firmly in Modernist thought, Hegelian triadic dialectic conflict, and Futurist social theory.
3. The Falangists sought to take the traditional Spanish monarchy and Catholic concepts and render them viable in the modern era through synthesis with their reaction (Enlightenment thought), to varying degrees of success. They were national syndicalists and civic nationalists who imposed Catholicism as the state religion but didn't care about racial purity or "The Jewish Question".