How would things play out if Kornilov's overthrow of the provisional government had been a failure?
No Russian Civil War, I'd imagine. Kornilov's coup was the main trigger for the outbreak of hostilities, although other factors were relevant; Kerensky's flight to Novgorod, as a direct result of this, sparked the fighting in Petrograd and the brief Bolshevik uprising. Without Kornilov, I'd say Kerensky's government would have managed to reestablish control over Russia eventually - there just weren't any credible opponents apart from the counterrevolutionaries, as the Bolsheviks and later the Socialist Revolutionaries proved, and Kornilov's coup was in some ways the last desperate throw of the dice for the counterrevolutionaries.
No Russian Civil War, in turn, presumably means that the country's a lot stronger in the early 1930s, and so doesn't crumple when the Alliance attacks it...
I'm doubtful Kerensky would manage to reestablish his authority; he had lost pretty much all of his credibility even before the Kornilov Coup. Instead, we'll probably see a Green or maybe even Red government take power. If the coup fails, Blue commanders like Kolchak, Yudenich, and Kornilov would probably flee to France and set up some rival government, though it won't have a prayer of reentering Russia.
No chance of a Red government - events proved quite conclusively the weaknesses of the Bolsheviks, who had a pretty good leader but not really anything else, and made far too many enemies to seem at all credible as a government. The SRs weren't much better - although they were fairly strong in the rural areas, they didn't have that all-important support of the cities, and their overall weakness was shown by their dramatic disintegration after the Battle of Tver - although this might be completely different without Kornilov's coup, since that had quite an impact on them.
Good point. Somebody has to establish a central authority in Russia, though. The Green SRs were IOTL not the strongest faction, but in this TM it seems like the situation is ripe for them to take over.