Yup, you've got it, that's exactly how it works for sparring. Not so much for the other types of training because they're more like tutorial sessions from the coach. There are lots of factors that affect training.
- your fighter's energy levels
- the quality of the coach
- the number of training partners in that session (the more there are the less he will improve because he won't get as much 1to1 coaching). this doesn't apply to sparring though as long as there is someone to spar with, you'll rotate through everyone sparring at that session.
- the gym quality (equipment in certain types of training)
To answer that one point specifically, it doesn't matter whether the training partners are your own fighters or someone elses.