The charge conjugation operator $C$ reverses the charge of a state. But it may or may not convert a particle to its antiparticle. For example, consider a neutrino which is charge-neutral and left-handed while its antiparticle is also charge-neutral but right-handed. Therefore, charge conjugation is not sufficient to produce the antineutrino from a neutrino but CP is. $P$ is responsible for changing the chirality.
My original answer was much too long, it didn't answer the question clearly (if at all), and it made a couple of dubious assertions. For those reasons, I replaced it with this new answer.
Answer to part 1
There is no natural pairing between particles and anti-particles at the level of individual states. There is only a natural pairing between particle species — that is, between species and their anti-species. That's because Poincaré transformations don't permute species, but they do permute states. This statement also applies to discrete Poincaré transformations like a space-reflection transformation P, in theories where P is a symmetry.
Once we agree that the appropriate pairing is between species, not between individual states, the answer to part 1 of the question is easy:
In a theory that has both CP and C symmetry, they both permute species the same way. It's not either-or, it's both.
To accommodate theories that don't have C symmetry, we can use CP to define the paring between species and their anti-species. This works whether or not the theory has C symmetry, and when it does, C symmetry gives the same pairing.
Even more generally, we could use CPT (instead of CP) to define the pairing between species and their anti-species, as Weinberg does in chapters 2 and 3 of Quantum Theory of Fields, volume 1. This provides some context for part 2 of the question...
Answer to part 2
Part 2 of the question asks whether part 1 is related to a couple of other questions about Sakharov's criteria for baryogenesis. Those criteria assert that to explain baryogenesis, the theory should not have either C or CP symmetry. The answer to
explains why this is so. To relate that answer to part 1 of the present question, use part 1 to see that Sakharov's criteria can be restated without referring specifically to CP or C, like this: To explain baryogenesis, the theory should not have any symmetries between particle species and their anti-species among all symmetries that preserve the time-orientation (which excludes CPT).