Kirk's going thru the KM test and already having the rank of Lt. is in perfect step with Saavik's experience at the start of Star Trek II.
And she also was apparently somehow still at the Academy, as was he.
That's a good point. Not every starfleet officer is going to serve on a starship and those that do especially command level probably have more training. Also in the case of McCoy, Starfleet Academy also serves as an officer candidate school for those with advance degrees (Kirk could have also been an officer candidate instead of a cadet)