No degree is required to become a college/university coach (although most have one in something); proven ability is required at that level. You start coaching at a lower level or as an assistant coach and when you prove you can consistently produce winners then you get offered a head coach position. When you prove you can produce winners as a head coach, better teams start trying to get you. When you're the very best, you get recruited by pros and the USOC. Nobody really cares about what degree you have or where you got it - only about how well you perform as a coach.
The only time the degrees really matter (except when the position also requires being a teacher) is when you have no coaching experience to back you up - in which case very few programs will be interested anyway unless they're desperate. Once you have coaching experience then it's the quality and level of that experience that matters.
There's a huge difference between a small NCAA-II school's periphery sports and a Division I big-time prime sport too. What's acceptable in some wouldn't get you on the field in some others. It varies widely from school-to-school and sport-to-sport.
That MS in Sports Psych won't be enough to get a job as a team psychologist, you'll need a PhD and a license from your state for that job. The MS Psychologist is usually a supervised assistant.
As for learning coaching skills via a degree; some majors are particularly useful. Human Biology is a good example. Kinesthetics is another. In fact, some colleges even have a master's degree in coaching theory. (example: http://cpass.wvu.edu/academic_programs/masters/athletic_coaching_education/online_information
Some coaching jobs in the university go to people that are qualified to also teach undergraduate courses (they often like to not hire more people than they really need) so an academic degree at the master's (minimum) or PhD level can be very helpful. Many of those jobs aren't faculty appointments though and go to the best qualified coach - not a teacher. They're also (except for many football and basketball positions) usually not full time.
is a pretty good place to start reading about the variety of requirements.