Has sexism really been an integral part of fascist propaganda throughout history?

What you have enclosed in your question is a list of "fascist minimums"—traits commonly associated with fascisms. This approach of studying fascisms has been criticized by political scientist and historian Robert O. Paxton, for falling into the trappings of creating a "medieval bestiary of fascisms".

Unfortunately much scholarly work on fascism complicates things still further* by two very widespread errors of approach. First, most authorities treat generic fascism in a static manner [...] — they look for a fixed essence: a famous "fascist minimum". Second, most works consider fascisms in too isolated a manner, without sufficient sustained reference to the political, social, and cultural spaces in which they navigate. Together, these two common errors of approach produce what we might call "bestiaries" of fascism. Like medieval naturalists, they present a catalog of portraits of one beast after another, each one portrayed against a bit of background scenery and identified by its external signs.^(1)

*Prior to this observation, Paxton lists five difficulties of identifying fascisms. These are:

A problem of timing (fascism was a sudden and unexpected phenomenon, resulting in a poor understanding of it)
A problem of mimicry (non-fascist parties adopting fascist traits in order to attract voters)
A problem of time and space (there is a great difference between individual cases)
A problem of ambiguity surrounding the relationship between doctrine and action in fascisms (fascisms are not rooted in any "Fascist Manifesto")
A problem of overuse of the term (everyone is someone's fascist)

The third and fourth problem listed above, I believe, are the most relevant to your question of sexism—or any fascist minimum, for that matter—being integral in fascist propaganda throughout history.

Needless to say, Paxton wishes to avoid falling into the trappings of the "Fascist Bestiary". He proposes we do this by studying fascisms as processes—developing through time and space, in a network of reciprocal relationships—and comparing them. He divides the fascist life-cycle into five categories. These are:

The initial creation of fascist movements
Their rooting as parties in a political system
The acquisition of power
The exercise of power
Radicalization or entropy

According to Paxton, comparison of stage 1 fascisms is of little help to us. Virtually all modern states have experienced proto-fascisms, because a prerequisite to fascism is a liberal democracy and a citizenry disillusioned with it. It is when a fascist movement attempts to root itself as a political party that comparison begins to be of help. This is because, firstly, not all fascist movements have been successful in this attempt, which should entice us to study and compare each individual case's circumstances. Which circumstances have enabled fascisms to thrive and root themselves as political parties, and which circumstances have disabled the same development? Paxton introduces two prerequisites for success. These are:

A weakened liberal democracy
A political deadlock between the Right and the Left

Success depends on certain relatively precise conditions: the weakness of a liberal state, whose inadequacies seem to condemn the nation to disorder, decline, or humiliation; and political deadlock because the Right, the heir to power but unable to continue to wield it alone, refuses to accept a growing Left as a legitimate governing partner. Some fascist leaders, in their turn, are willing to reposition their movements in alliances with these frightened conservatives, a step that pays handsomely in political power, at the cost of disaffection among some of the early antibourgeois militants.^(1)

I wish to accentuate the last sentence of Paxton's observations above, because this implies an abandonment of doctrine—showing how malleable fascisms can be.

Therefore, the methods of intellectual history* become much less helpful beyond the first stage in the fascist cycle. Every fascist movement that has rooted itself successfully as a major political contender, thereby approaching power, has betrayed its initial antibourgeois and anticapitalist program.^(1)

*The approach of studying fascisms in a static manner, usually resulting in a fascist minimum.

This becomes even more evident as fascisms acquire power. Paxton reminds us that no fascism has acquired its nation's power by coup d'état.

Fascism cannot appeal to the street without risking a confrontation with future allies—the army and the police—without whom it will not be able to pursue its expansionist goals. Indeed, fascist coup attempts have commonly led to military dictatorships rather than to fascist power [...] The only route to power available to fascists passes through cooperation with conservative elites. The most important variables, therefore, are the conservative elites' willingness to work with the fascists and the depth of the crisis that induces them to cooperate.^(1)

The last stage of fascism is heavily dependent on the fourth stage—the exercise of power. This is due to the reciprocal relationship agreed upon by the conservative Right and the fascists, and the four-way struggle described below.

Fascist leaders who have reached power, historically, have been condemned to govern in association with the conservative elites who had opened the gates to them. This sets up a four-way struggle for dominance among the leader, his party (whose militants clamor for jobs, perquisites, expansionist adventures, and the fulfillment of elements of the early radical program), the regular state functionaries such as police commanders and magistrates, and the traditional elites—churches, the army, the professions, and business leaders. This four-way tension is what gives fascist rule its characteristic blend of febrile activism and shapelessness.

Nazi Germany, contrary to Mussolini's fascist Italy, radicalized due to a strong party dominating the traditional state.

I have yet to answer your question directly, and therefore you might question the relevancy of my answer. My purpose, however, hasn't been to answer your question directly. Rather, my purpose has been to reveal, using Paxton's work, the weakness of the fascist minimums. It has been to reveal the malleability and shapelessness of fascisms—that once a fascism roots itself as a political party, it necessarily abandons parts of its earlier doctrine. This development continues into the later stages, as fascist exercise of power is constricted by vines of reciprocal alliances, upon which fascisms depend on for power. Paxton concludes that we should rather define fascisms by their functions, than their traits. A function of fascism would be to enforce political authority and reinforce unity. The manner in which a fascism enforces political authority and reinforces unity—the execution of its function—however, can be vastly different. A modern fascism in Europe today might be anti-Islamic, rather than anti-Semitic. A modern fascism might reinforce unity through feminism, contradicting traditional traits of the inter-war fascist leader, such as virility and machismo.

^(1) Paxton, Robert O. "The Five Stages of Fascism". In The Journal of Modern History, Vol. 70. No. 1 (March 1998).