I assume here that you're talking about the connection between Futurism and Fascism in Italy specifically. Futurism as a movement was not fascist in origin. By the time the Fascists came to power, there was some overlap in ideology when it came to visions of a future society and so forth, but Futurism is not inherently linked to a Fascism.

The futurists (Mayakovsky et al.) in Russia were supported by the early Soviet government and similarly looked at the revolution with hope. In the Soviet Union they were before too long dismissed as formalists (as were many other avant garde artists/art forms in the late 1920s early 30s in the Soviet Union). A similar thing happened in Italy in the late 1930s.

So, to address your points individually. No, it does not have its very origins in Fascist propaganda. Also, Futurism was not monolithic enough to say that it was "transformed and used by Fascists to promote their agenda." Even if we are talking solely about Italy, the connection between the two is a bit more complicated.

No. The first Futurist manifesto (Marinetti's The Futurist Manifesto, published in 1909), while having proto-fascist ideas such as "[w]e want to glorify war — the only cure for the world — militarism, patriotism, the destructive gesture of the anarchists, the beautiful ideas which kill, and contempt for woman", was not tied to any specific political party or ideology. Marinetti was trying to imagine an idealized, violent, mechanical future that he thought was inevitable. Carlo Carra, another early Futurist member, supported anarchism (evident in his painting Funeral of the Anarchist Galli). Futurism began strictly as an art movement that appropriated ideologies like socialism and anarchism because of their call to destroy bourgeois culture, and their provocative, propagandistic style was in the beginning strictly aesthetic.

Futurism wasn't co-opted by Mussolini's National Fascist Party, but instead, a few years after he founded the movement, Marinetti tried to make Futurism the official art of Fascist Italy, although he never succeeded and eventually condemned the NFP. The Futurist style, though, the use of manifestos especially, influenced a lot of propagandists and found itself being used for both far-right and far-left ends.

Did Fascist governments utilize the futurist style more than Socialist or Communist governments? Even in the Capitalist West, government projects like the American WPA used futurist-inspired art to promote "American values" and raise morale.