So I think that first part depends, and not sure if we can get a direct answer for that, beyond specific individuals. Overall, I think professional boxers are probably more competitive than most people, so they may be more likely to train for competitions, but at the same time I can see them view it as simply a tool, and not care about competing with it. So different boxers different answers.
As for the athlete thing-I don't really know what point you're making with that. The fact that some people can/do switch between them, doesn't mean it's the most effective route to do it. And with MMA and BJJ I think that might be an exception to the idea, considering just how similar they are-nearly 100% of BJJ can be transferred directly into MMA, so for one-direction there is nothing being lost. And then there are other people who aren't high-level BJJers (meaning they don't enter high-level BJJ comps) that are high level MMA fighters that are also successful, so it doesn't suggest one method is more effective than the other.