The whole game is a direct consequence of Crono&co's actions. How can it not be a sequel? Because it follows a different cast?

But sure, if you go in expecting to see more of Crono's silly adventures in time, you'll get disappointed. Expect another great story in the Chrono universe however, and you'll be rewarded.

The C&C Encyclopedia is a much better resource.

It doesn't matter which ending you take, the Soviet Union falls either way. Basically, the Soviet campaign just provides background information (the Nod stuff). Therefore:
RA -> C&C -> TS -> Firestorm -> C&C3
(RA ->) RA2 -> YR