If a probe were sent to another solar system, could it calculate and insert itself in orbit around a planet?

Well, in theory, it is the same concept as getting a probe in orbit around a planet in our solar system. Of course, over such long distances (and long, long, long times), a computer on board the probe would have to be able to do all of the calculations. I don't know that we have the technology for the probe to do its own calculations without human intervention, but I'd say its possible. Don't hold your breath.