The Terran backstory (animation scenes) in Starcraft has multiple Russian Generals. Why? Did that arm of Terrans evolve directly from the U.S.S.R?
There are 3 main terran factions that appear in the original game. The Terran Confederacy, The Sons of Korhal, and The UED (United Earth Directorate). Only the UED have the Russian Generals that you mentioned. And they make a point in the game to say that UED are the only Terran fleet directly from earth. It might stand to reason that Earth is now a USSR modeled form of government (or the UED are based out of Russia), but not all Terran forces follow this model.
The Terran came from Earth but got lost in space. With their existing knowledge, they formed governments and a military. Then there was a revolution turning the government into a dictatorship.
First of all the Terran's didn't evolve from anything from earth. They are from Earth, in a matter of speaking. Their "race" is from Earth. As you see from different characters, like Duran, Raynor, Kerrigan, Mengsk,.
Blizzard probably made the Generals look Russian to make a reference that the Soviets have a dictatorship, just as Mensk is the Dictator of the Terran Alliance.
Short answer: no.
In the original game manual, pages 26-27, there's a description of Earth history leading up to the launch of the sleeper ships (emphasis mine):
By the end of the twenty-first century, mankind had seen bold and unprecedented changes within the world. Radical new technologies were surfacing at incredible rates.... In the wake of the eradication of Communism from the Eastern Nations, nuclear weapons quickly became available in abundance. The international power structure... was blasted apart as third-world nations rose to challenge [the superpowers].
Popular sentiment held that the world was plummeting towards an inevitable catastrophe as overpopulation and genetic alteration swept across the globe.... Horrific acts of terrorism and violence erupted between the corporate sector and the humanist factions, resulting in forced police actions across the globe.... Ultimately, the precarious balance of world power exploded into international pandemonium.
On November 22, 2229, the United Powers League was founded. The UPL was to become the ultimate incarnation of the vision of a unified humanity held by the now-defunct United Nations.... The UPL was founded on the basis of "enlightened socialism", but often resorted to harsh, fascist police actions to maintain public order.
From this, it seems unambiguous that the old social order, nations and all, had essentially fallen apart by the time the UPL came around, with special emphasis on the collapse of Communism. The UPL itself seems to have foundational ethics from all over the place; certainly it has some Stalinist elements, some Communist ideas, but it isn't a direct descendant of any one single Earth ideology or nation.
When they realized that the Terrans in the Koprulu sector had discovered intelligent alien life, the UPL was reorganized into the UED with the express aim of defending Earth by any means necessary (read: purge all the xenos).
The only characters in StarCraft with a Russian accent is Alexei Stukov, one of the leaders of the United Earth Directorate invasion from Earth to the Koprulu sector. Stukov, due to his accent, can be assumed to have had his origins from the old Russia on Earth.
That does not mean that Earth was governed by Russia. Actually, in the StarCraft lore, it is stated that the story departs from a point where the USSR never broke. It's vaguely explained that The world politics made some twists based on some new religion, related with pure-human (humans with no implants and so), supremacy.
Criminals and outcasts were sent in 4 big ships as a colonial experiment, and some of them survived, founding 3 colonies in different systems of the Koprulu sector.
The game lore starts a few hundred years later.