The rehiring of Coaches is not all that hard. Just click on the ones you want to keep. The frustrating part in most cases is the ones that want to leave is the ones you would like to keep. Keep in mind this is a game of ratings and as well as players, coaches are also governed by ratings. You can also promote coaches at the same time.
Bench Coaches are more important at the minor league levels than the ML level. They should have a high rating in Strategy. The Bench Coach at the ML level takes over when you get tossed from the game, it does happen every now and then (or more in some cases).
Pitching and Bull Pen Coaches need a high rating in the Pitching IQ.
Hitting Coach needs a high rating in the Hitting IQ.
Base Coaches need a high rating in the Baserunning IQ
Fielding Instructors needs a high rating in the Fielding IQ.
The Patience rating is more important for minor league coaches unless you have a young team at the ML level. The Discipline rating according to the players guide should be high but I am not real sure that it has all that much bearing. A high Loyalty rating means the coach is more likely to return for further seasons, I wouldn't bank on it.
Rehiring is easy, wait until it is hiring time to fill empty coaching slots. The hiring phase is worse than going after a Free Agent.