Timo said:
Regarding Crusher's qualifications, it's highly unlikely that she would be a line officer in any sense of the expression. Judging by "Thine Own Self" dialogue, the training and test she took qualified her to "stand bridge watches", to be Officer of the Deck whenever the Captain so decrees - a no-brainer job in today's navies that is usually given to the most junior officers available. In essence, the test probably made Commander Crusher equally eligible for OOD duty as the lowliest yellow- or redshirt aboard. Which is a handy feature, gotta admit.
Nor does it seem that such training is usual among medical personnel, since neither McCoy nor Pulaski ever displayed such qualifications. The superman Bashir no doubt eagerly volunteered for training of this sort early in his career, but McCoy and Pulaski seem to have made it to high rank without ever considering command tasks. (Command in ship operations, that is - they obviously command their respective medical staffs.) The command test must be but one out of dozens of different means for gaining enough brownie points for a promotion.
As for the logic of Picard leaving her in command... Well, that would be a better-protected assignment than any of the surface teams. Picard might face some tough questioning for such a personally driven decision, but from his viewpoint it must have been rather sound. Keep Beverly out of danger, and keep her where she can quickly deploy to any planetary spot that requires her medical attention. The one thing Picard really should have done differently is make her remain aboard as the CMO, not as the OOD!
Timo Saloniemi