What if Italy goes Communist in the 20's, beating the Fascists and overthrowing the King?
mmmh, interesting idea, but unlikely.
Italy was thorn by strikes and other social issues (economical crisis, veterans still unemployed, almost feudal agriculture and social structure in the south), still worker class just did not had the power or the instruments (cultural and material) to start a revolution.
the same fascist movement and the march on Rome was seen as (and it was) just a minor public order nuisance, nothing that one or two regiments of Carabinieri Reali could not take care of in an afternoon. it was a surprise for a lot of people that the king actually received and spoke with Mussolini instead of having him and his bunch of friends arrested.
that is to say that the government would still have a lot of power.
keep also in mind the difference between the relatively progredite, industrial north and the arretrate, agricultural south: it would have been hard to find some common ground between the proletarian of this zones.
there were intellectual and agitators but there were also a lot of analfabetism, so circulation of new ideas was slow. peasants in the south colud easily be intimdated by big land-owners.
Who would lead a Communist Italy? Who were the prominent Italian Communists at that time? What brand of Communism could come out of Italy? How would Italy do economicaly and militarily?
for what concern the main intellectuals, I'll write down some name, but then you should make some research for your own, because as much as this thread could be interesting (I'm italian, by the way) unfortunately I'm not such a great scholar.
you could concentrate on Antonio Gramsci, Palmiro Togliatti(two "orthodoxes") and Amadeo Bordiga (somewhat more interesting ideologically, even if always minoritarian), Umberto Terracini. anyway the socialists mantained a good position, so you may search also on Nenni and Turati.
Who would lead? given that a communist coup in Italy at the time would have impossible (in my opinion) without external aid, and given that the only source for such help would have come from the USSR, and given that the most USSR-aligned were Togliatti and Gramsci (Bordiga was dismissed ad "sectarian" and "trotzkist"), I'd say Togliatti.
Gramsci was surely a significative personality, but he was more an intellectual devoted to the education of the masses than to power games, and was also of weak health. His ideal place would have been minister of culture and propaganda or something like that. interestingly, some intellectual of the contemporary right wing in Italy have a good opinion of him.
about the brand of the italian communism. well, post-war italian communist party was the largest in europe because it was the less communist. also, it contributed to the liberation from nazifascism with other political forces and to the founding of the modern italian state and its Constitution. it has to be said that it was Togliatti (then head of the party) that pushed for a legalitarian presence of the communist party inside the institutions and abandoned any velleitarian ideas of armed insurgence (if not to counter an always possible rightwing-neofascist coup).
but this was post-WW2.
to return to the '20's, I'd say that the revolutionaries should take care of the bipolar structure of Italy: industrialized north, agricultural south.
to begin with the south, it's probable that the communists would have tried to involve the population in some sort of guerrilla, as it happened with the "briganti" (bush bandits) when the Savoy kingdom (north) invaded the Borbone's Two Sicily kingdom (south). The briganti were sometimes organized and leaded by borboneses nostalgic army officers.
this time the leadership would have cared also for agricultural organization, to provide food for the rebels. the south is poor, but if adequately used, the land could sustain decently a lot of people.
it's likely that if the peasants were organized in small communes and were granted decent life condition they would have provided a good agricultural workforce and production. after all, ok, we take away a lot from you, but is always less that your precedent masters took, and this time we'll use it to feed yous, ours brothers fighting for peace, welfare and freedom, or something like that. we teach to read and write to your children, we give adequate food to them etc., so for the first moment, at last, the revolution could take in the south.
the land owners could make up some armed men to counter the rebellion, but with a good organization this could be managed by the insurgents.
it would be a good politic not to clash with the church. with some ideological pragmatism and elasticity this would not be a problem.
all this consideration lead to a sort of paternalistical benevolent, I'd say bucholical, regime, in which the revolution could consolidate and the masses been educated to further revolutionary development.
in the north there would be more problem. there the situation is more close to the marxist theories, there is a capitalist industrial structure ripe for revolution, but there is still a lot of people who have a lot more to loose than their chains from a communist revolution, so the resistance would be strong.
the worker class still has no culture enough to run the industrial plants for their own.
memories are still fresh of the foreigner domination, ended only with the second world war, so any interference from the outside would not be tolerated easily.
What relationship would it have with the Soviet Union? How would neighboring countries and the West react to a Communist Italy?
well. let's say italian communists could use weapons from USSR. surely there would be some ideological clash about the "italianization" of communism I have hypotized before, but if this is taken with pragmatism and if Togliatti is good enough to coinvince his idol Stalin that is just a matter of time before getting more orthodox (and of course if the same Togliatti first agree on this) then they could go along well. Southern Italy could become a second and more pleasant Crimea in which to build dacias and sip wine, a soviet Cote Azure.
the neghboring countries (france, austria, yugoslavia) would be enough in their own post-war and economical trouble to care for what happen in Italy, and they border on the north, where the "revolution" would be harder to realize. switzerland just would not care. germany is in even bigger troubles.
maybe UK would be concerned about soviet fleeing in the mediterranean, but UK is far away.
on the other hand, after a while bourgeois capitalist nations could be strong enough to be concerned about it. remember that Italy even if formally united under a communist regime would be still a split country: real communism north, agricultural communism in the south, with a lot of discontent in both places. it could be easy for an alliance of foreign nation to take control of Italy (ad it happened for centuries) or at least try.
given that one cannot see an italian Generalissimo Franco, even if italian collaborate to the "liberation" from the communist, it's likely that the result would be regional states ruled by puppets of foreign nation. maybe they will also tolerate a minoritarian communist regional state somewhere.
Is there a chance for Communist Italy to become more than the second rate power, Fascist Italy was in OTL? And what if it did?
no chance. if it's lucky enough WW2 does not take place, or at last stay neutral, or if the nazi still develop in germany, Italy will start the war this time from the "right" side since the beginning, and eventually is "liberated" from the USSR as well.
Italy would not develop a decent industry under the "guidance" of the soviet, and it would be lucky enough if they don't put lisenkian hands in southern agriculture.