I've noticed with today's patch that they those listed sometimes appear to have multiple jobs.

For example, a Security Officer would also be listed as Hazzard and Astrometics. So I suspect this could be the system picking up similar names from the different professions.

not only that, but also I don;t understand why Klingon DOffs are on the Fed Exchange and vice versa, why if we can't use them?

For the same reason that you can pick up fighters and KDF BO's fed side too: The Exchange wasnt build for handeling faction specific content.