All the officers are potentially action characters because they are trained officers. The trick is using them in scenarios where they are useful and not shoe-horning them in.. So no, Uhura should not be sent on a mission instead of security staff (unless as the officer in charge) but yes, she should be sent on a mission where her skills (computer expertise and fluent Romulan) might be of vital importance.
Right. The trick is not to put them into a situation that seems unbelievable for that character and his or her position and expertise. Giving McCoy a phaser and telling him to go shoot some Klingons doesn't make sense. Or, suiting him up to space jump down to a Romulan drilling rig would be ludicrous.
That said, McCoy certainly seems to have an action scene in the first nine minutes of the movie, but it's also one more suited to his character and wasn't intended to turn out as it did. But, I'd expect to see Sulu or Chekov, or even Uhura on Kronos with Kirk and Spock before McCoy.
Parenthetically, am I right to think that the only time we saw McCoy fire a weapon was when he killed the salt monster in "The Mantrap"?