I have a CSCS and my opinion of the test is this:
It's a fairly difficult test, but if you study their material you should pass both sections on your first try. A lot of people don't pass both at once though. That was apparent because when I took the test there were many people that took one section but not the other, so you could tell that they had already passed one part but needed to make up the other.
I took the test about 8 years ago and even at that time, I thought that some of the material was out of date. I have no idea if they've updated the test or not. Hopefully they have, but in any event you need to take it if you really want to be an S&C coach as many coaches passed that exam.
In regard to your degree, you need a Kinesiology degree or an exercise science degree. An athletic trainer degree will train you for dealing with injuries, while a health/fitness type degree will sometimes be looked down upon by coaches as fitness is really a different field.
The most important things you will need are experience and contacts. So you must volunteer as an intern in your schools strength and conditioning department and do a great job. While you're at it try to go to as many seminars and conferences that you can and make as many contacts as possible. It's a very insular field and even when jobs look like they're open, the people doing the hiring usually have someone in mind to take the job before the position is even advertised.
Hope that helps.