There is always something more to be said, but this wonderful answer from 2014 by /u/depanneur could be of use. Of course, not all of your subquestions are addressed, but it's a start!

There is a plethora of definitions for Fascism, from defining it as a very narrowly by limiting it to the historical phenomenon of Mussolini's rule in Italy to a very broad definition like the one used historically by Marxists, which I have described in-depth in this thread.

The one I found most useful and sensible within the context of my own historical work is the approach used by Robert Paxton in his book The Anatomy of Fascism, which defines Fascism from a praxeological standpoint.

Paxton points out in his introduction that despite Fascism being a major phenomenon of the 20th century, even now no definition of Fascism has obtained universal assent as a complete satisfactory account of the phenomenon. Fascist movements varied so strongly from one national setting to another that some scholars even cast doubt that the term is more than a political smear word.

However, it is also impossible to ignore how many movements in inter-war Europe and even beyond chose the descriptor of Fascism for themselves as well as what kind of structural and practical similarities existed between many of these movements.

One of the major factors, Paxton points to when examining Fascist movements its view on what drives history: Unlike the advocators of liberal democracy, it is not reason or modernization, which drives forward and unlike communists, it is not material relationships. For the Fascist the engine of history is conflict, whether between nations, peoples or races. History is a constant struggle in which a community of mythical qualities needs to assert itself in order to gain dominance over others. Dominance is the core goal and must be asserted. And only if the right and rightful people dominate will a golden age begin.

The political utopia of the Fascist differs greatly from liberal or communist visions of utopia: Both of the latter are build on a vision of a utopian future that needs to be build and achieved. The Fascist on the other hand looks to the past for its utopia since most fantasies of dominance are historically justified. Whether it is the return to the Roman Empire or the mythical Lebensraum of German kings, all Fascist utopian visions are build upon a return to a hazy, mythological past in which the world was right.

This factory is depended on and at the same time leads to the strong inherently anti-modernist rhetoric of Fascist movements. From tropes such as the city corrupting the purity of rural live or the return to a blood-and-soil type romantic idyll or decrying the devaluation of the core family and the place of women in society as "unnatural" Fascism espouses a rhetoric that uses the past to justify a complete and total criticism of the present and advocates building an alternate modernity in the image of a supposedly "pure" past.

Consequently, Fascists see themselves as not merely espousing an ideology but rather, a creed or – as the Nazis called it – a Weltanschauung (roughly translated as "world view"). As Paxton writes:

In a way unlike the classical "isms", the rightness of fascism does not depend on the truth of any proposition advanced in its name. Fascism is "true" insofar as it helps fulfill the destiny of a chosen race or people or blood, locked with other peoples in a Darwinian struggle, and not in the light of some abstract universal reason. (...) The truth was whatever permitted the new fascist man (and woman) to dominate others, and whatever made the chosen people triumph.

Fascism rested not upon the truth of its doctrine but upon the leader's mystical union with historic destiny of his people. (...) The fascist leader wanted to bring his people into a higher realm of politics that they would experience sensually: the warmth of belonging to a race now fully aware of its identity, historic destiny, and power; the excitement of participating in a vast collective enterprise; the gratification of submerging oneself in a wave of shared feelings, and of sacrificing one's petty concerns for the group's good; and the thrill of domination. Fascism deliberate replacement of reasoned debate with immediate sensual experience transformed politics, as the exiled cultural critic Walter Benjamin was first to point, into aesthetics. And the ultimate fascist aesthetic experience, Benjamin warned in 1936, was war.

Fascist leaders made no secret of having no program. (...) Fascism radical instrumentalization of truth explains why fascists never bothered to write any casuistical literature when they changed their program as they did often and without compunction. Stalin was forever writing to prove that his policies accorded somehow with the principles of Marx and Lenin; Hitler and Mussolini never bothered with any such theoretical justification.

In the same vein, Paxton goes on to define Fascism as

a form of political behavior marked by obsessive preoccupation with community decline, humiliation, or victim-hood and by compensatory cults of unity, energy, and purity, in which a mass-based party of committed nationalist militants, working in uneasy but effective collaboration with traditional elites, abandons democratic liberties and pursues with redemptive violence and without ethical or legal restraints goals of internal cleansing and external expansion.

What is important here is not only that Fascism operates always under the assumption of threat resulting from the inherent view on the way history always functions as a conflict; it's also imperative that Paxton defines it as a form of political behavior rather than a stringent world-view.

In service of the core narrative of threat and conflict, an ill-defined and mythical past to which the goal is to return, and Fascisms use of authoritarian means to these ends, Fascism is the turning of politics into spectacle and an aesthetic experience. It doesn't matter in a sense what kind of program it is espousing at the moment but the imperative lies in an utopia with an open definition in whose service the experience of community against the forces that threaten the community, the way of life, the "chosen race" have rallied. Who these enemies are depend on historical and national context, it is the underlying practice that is so essential to the political behavior of Fascism.

Paxton's approach to the subject has been immensely useful in my own studies of Nazi Germany because rather than listicles providing definitions of fascism according to its content or trying to cut through the literature published by Nazis and Italian and other Fascists in order to analyze common threads and construct a Weberian "ideal type" of fascism, this approach is best able to serve as a framework when it comes to what is often perceived as the internal inconsistencies of Nazism, from revolutionary rhetoric in its early periods to alliances with traditional social elites to the eclectic mix of Roman, Germanian and other aesthetic to serve propagandist ends. Even the self-description of "National Socialism" can best understood as embracing a certain political methodology and practice rather than a profession for what we tend to understand as any form of actual socialism.

Fascism is therefore about function and form, not so much concrete content. Paxton goes on in his book to examine historically fascist movements throughout the various stages they moved through and while expanding upon them here would probably lead too far, I can not recommend his book highly enough. Towards the end of his book, he writes about Fascism outside of Europe and in light of recent renewed interest in the subject in the US, I think the following passage is especially pertinent:

The United States itself has never been exempt from fascism. (...) Much more dangerous [than movements like the American Nazi Party, which utilize already established tenants and creeds from Europe] are movements that employ authentically American themes in ways that resemble fascism functionally. The Klan revived in the 1920s, took on virulent anti-Semitism, and spread to the cities of the Middle West. In the 1930s, Father Charles E. Coughlin gathered a radio audience estimated at forty million around anti-communist, anti-Wall Street, pro-soft money, and – after 1938 – anti-Semtic message broadcast from his church in the outskirts of Detroit. (...) Today a "politics of resentment" rooted in authentic American piety and nativism sometimes leads to violence against some of the very same "internal enemies" once targeted by the Nazis, such as homosexuals and defenders of abortion rights.

Of course the United Staes would have to suffer catastrophic setbacks and polarization for these fringe groups to find powerful allies and enter the mainstream. I half expected to see emergence after 1968 a movement of national reunification, regeneration, and purification directed against hirsute antiwar protesters, black radicals, and "degenerate" artists. I thought that some of the Vietnam veterans might form analogs to the Freikorps of 1919 Germany and Itlaian Arditi, and attack youths whose demonstrations on the steps of the Pentagon had "stabbed them in the back". Fortunately, I was wrong (so far). (...)

The language and symbols of an authentic American fascism would, of course, have little to do with the original European models. They would have to be familiar and reassuring to loyal Americans as the language and symbols of the original fascism were familiar and reassuring to many Italians and Germans. (...) No swastikas in an American fascism, but Stars and Stripes (or Stars and Bars) and Christian corsses. No fascist salute, but mass recitation of the pledge of allegiance. These symbols contain no whiff of fascism in themselves, of course, but an American fascism would transform them into obligatory litmus tests for detecting the internal enemy."

Asking an additional question, how much does Trotsky’s analysis of the class basis of fascism get looked at by modern historians?

EDIT: I asked the question

Obviously there are many, many definitions of fascism floating out there, some better and more scholarly than others. While u/commiespaceinvader's comment does a great job explaining the Robert Paxton definition of fascism, I think it would be useful to discuss other leading alternative definitions, particularly those that emphasize ideology over function.

When commiespaceinvader refers to some definitions "trying to cut through the literature published by Nazis and Italian and other Fascists in order to analyze common threads and construct a Weberian "ideal type" of fascism", he's referring to the Roger Griffin definition:

"Fascism is a political ideology whose mythic core in its various permutations is a palingenetic form of populist ultra-nationalism."

Or, put without jargon, fascism is at its core a form of ultranationalism that, through the use of utopian mythology (be it existing national myths or newly constructed narratives), seeks to generate a rebirth (i.e. "palingenesis") of the "ultra-nation" (i.e. an organic people plagued by decadency and general political/moral crisis) through revolutionary and populist means. As commiespaceinvader noted, Paxton's definition focuses on behaviour, forms, and functions of fascist parties in practice; Griffin, on the other hand, looks at the ideological content of fascism to distill a minimum definition--that is, what the textbook-ideal fascist "true believer" would hold.

So to answer OP's questions (I don't have his books on hand at the moment), I think Griffin would say that it's the rhetoric of, and nominal commitment to, the idea of radical national rebirth as *the * central mobilizing objective of society that distinguishes fascists from other forms of dictatorships. Although fascists may mine a mythic past to develop their narrative of national decline, they are products of modernity in the sense that they desire to remake society wholesale. Fascists must also be revolutionary, in the sense that they must be committed to dismantling democratic institutions and the existing order, no matter how they got into power. Griffin's definition prescribes no fascist programme for things like economics, other than the fact that economics is subordinated to the ultimate objective of national regeneration.

However, the sociological form, function, and trajectory to power of fascist groups (from vanguard party to terrorist cells) don't matter as much because, for Griffin, fascism is a virulent, "protean" idea that is adaptable to new circumstances and historical contexts. Even within interwar fascism, Griffin pointed out that there were significant divergences between regimes and movements; his definition is a basic narrative/script the contents of which vary depending on a country's unique national history and culture, if not between internal factions within a single party/movement.

Of course, postwar fascism is a whole 'nother kettle of fish. Postwar fascists operate in an environment where WWII's shadow looms greatly, where liberal democracy was virtually unassailable. For these, Griffin noted in his earlier 1991 book that there were basically three types of postwar fascism (that dominated at least through the Cold War):

1. Nostalgic fascism (i.e. surviving remnants of interwar regime personnel and organizations, like Waffen SS veteran groups or the first generation of parties like the Italian MSI or Austrian FPO)

2. Mimetic fascism (i.e. social outcasts adopting fascist and Nazi semiotics to express grievances, like the British National Front or racist skinhead bands)

3. Neo-fascism (i.e. new, genuinely innovative developments in far-right intellectual development, such as the French Nouvelle Droite)

I think here you can see the main difference/contribution between Griffin and Paxton: Paxton's definition doesn't have space to discuss non-party movements or milieus that have no chance or intention to seize political power themselves. (Griffin's commented on more recent manifestations of fascism like the terrorism turn and the alt-right, but that's another discussion; I'm respecting the 20-year rule)

Like I said at the offset, there are too many definitions of fascism floating around, and some are more useful in certain contexts than others. I know commiespaceinvader is a Holocaust/WWII historian so Paxton's definition works much better for his work since Paxton's frame of reference is based on interwar party movements, but Griffin's definition is more conducive for viewing fascism across time and space, before and after WWII, since he prioritizes fascism as an intellectual ideology rather than a set of behaviours.

If you worry that Griffin's definition isn't specific enough, I will throw in one other definition, closely aligned to Griffin's, by Roger Eatwell (a political scientist with a historical focus). He developed a definitional "matrix" for fascism. To be considered a fascist, you have to subscribe to three themes:

1. A holistic nation: "Fascists promise to forge a spiritual community that demands total loyalty and devotion to its interests"

2. New Man: "Fascists promise to create a communal and spiritual 'New Man' under the direction of dynamic new leaders"

3. An authoritarian third way: "Fascism promises to create an authoritarian state-led socioeconomic third way between capitalism and socialism"

For Eatwell, fascism's demands therefore are far more ambitious and likewise revolutionary than, say, populists (who merely promise to represent the existing popular will and remove corrupt elites; again, won't go any deeper than that).

Suffice it to say that, for both Griffin and Eatwell, one can only understand fascists by understanding their political objectives and recognizing that they're driven by some form of revolutionary ultranationalism with ambitions to remake human nature if need be; their political behaviour and tactics and strategies, whatever those may be, would flow from, and in serve of, this foundational principle.

Sources:

• The Nature of Fascism (1991) by Roger Griffin
• Fascism: An Introduction to Comparative Fascist Studies (2018), by Roger Griffin
• National Populism: The Revolt Against Liberal Democracy (2018), by Roger Eatwell and Matthew Goodwin