Hey, guys. A lot of these questions venture into questions about our methods that I won't answer, but I'll try to get to some of the others.
Besides using a time machine to go back and study mathematics/statistics or law, what advice do you have for anyone interested in pursuing your career?
I don't know your situation, but in general, people who do analytical work for teams are people who have spent many years doing baseball research in their spare time because they love figuring out how baseball works and can't help but be constantly digging into the data to see what answers they can find. (The exception to that would be for database gurus. If you can be the database architect, you might not have to be inquisitive about baseball.)
If you are that type of person, here are some additional qualifications that are helpful, roughly in order of importance:
1. Communication and interpersonal skills (it doesn't do any good to have ideas if you can't communicate them effectively and aren't a good listener; a lot of people in this role will be nerdy, but it's a huge help if you can talk with coaches and scouts in their language)
2. Research skills, good logical thinker and problem solver (there are a lot more people who do bad baseball research than those who do it well)
3. Database skills
4. Baseball knowledge, including video work, scouting knowledge, etc.
5. Web development skills
6. Statistical skills
7. Business acumen
8. Leadership and people management skills