In order to follow the NCAA rules, the parent has to be employed for 5 years before the athlete can take advantage of this.
Yes, though each school has their own policy regarding how long an employee must work at the school in order to be vested enough to receive tuition remission for family members. In my experience in working at a few universities, I've seen it where a school allows 100% tuition remission immediately upon hire, another where you are gradually vested (e.g., 25% tuition remission after 2 years, 50% after 3 years, 100% after 5 years) and one where you are only fully vested after 8 years working at the school. I don't know UNL's policy. Good question though.
No. Not at NU, and it would violate NCAA rules to boot. See here:
Eligibility for Dependent Scholarship Program
As noted above, however, you have to consider NCAA regulations on this matter as well, not just the university's policy.
As pointed out, NU only gives you 15 credit hours per year which does not include any fees, books, board, etc. which can be given to a spouse or dependent. However, under NCAA rules if the coach has not been with the school for 5 years, that counts as institutional financial aid and "counts" toward the 85 limit. He would also apparently not get all of the extensive benefits (including training table meals) you are allowed as a full scholarship student athlete. But on the plus side: "An institution may provide fruit, nuts and bagels to a student-athlete at any time."
Cotton's prior time spent here may allow him to appeal to the NCAA, but he still would only get 50% of tuition for one of his sons unless his wife is working for the U. Regardless, I think he'll turn out to be a good player.