Is every single BO ability available for purchase from some trainer somewhere in the game? Or are there certain abilities that one will only ever find on rewarded officer candidates? If the latter is the case, how can I find out what they are?
You can only train level 1 and 2 skills from the trainer. Level 3 skills must be gained from other bridge officers or by unlocking the skill through your skill tree. Oh, and they're not all Captain level skills. For instance, High Yield 3 trains from Starship Torpedo Weapons. There are Level 3 skills all over the tree, from Lieutenant Commander up to Admiral. Some are better than others. Some aren't worth getting. For instance, do you really wanna waste Admiral points in Astrometrics to get Particle Burst 3? Didn't think so.
Not trainable are a couple of BO and kits abilities:
Transphasic shield (or was it the bomb?)
and some more.
I'm not entirely sure about Rally Cry and Antimatter Spread, but Transphasic Bomb and Triage are definitely ground kit abilities for player characters that CANNOT be used by BOs. They're kit powers for players only.
A similar situation applies to the powers you get in space. Engineer players get, say...Rotate Shield Frequency, EPS Power Transfer, and so on in space. You'll never find a BO with these powers. They're player powers only.