1. They're going to school to play soccer.
2. Correct (Also hilarious)
3. "Both you and Cindy are getting great jobs" is fine. The "Both" in not needed however, so "You and Cindy are getting great jobs" would be better.
4. This one is very tricky. Technically, "Neither you nor I am Mr. Lin's student" would be correct, because the verb is closest to "I." Most people would not say this though, because it sounds terrible. A much better way to say this would be "Neither of us is Mr. Lin's student."
5. "You and John are not good students." "You and John are good students." You can but both in front if you want, but it's not necessary.
6. Both he and she are doctors. A good way to remember this is make it two sentences and see if it works. so "He is a doctor" and "Her is a doctor" doesn't work, but "He is a doctor" and "She is a doctor." does.