1. Technically, Chancellor Ronnie Green is the authority over the AD, he fired him. According to the University bylaws the Chancellor handles: "Personnel administration, including employment, termination, wage determinations, and conditions of employment for other than academic-administrative employees of the major administrative unit, in conformity with University-wide policies, practices, and procedures as may be required by the Board (of regents)"
The President is in charge of the four Chancellors for UNL, UNO, UK, UMC. A "major administrative officer" for UNL is nominated by the Chancellor to the President. If the President approves, the nomination is submitted to the regents. if they approve the nomination, the Chancellor is (technically) in control of that person. There is nothing in the bylaws that allows the President to fire an administrative officer. On the other hand, there is nothing that indicates he/she can't suggest the Chancellor fires someone. Same for the Regents, they can't fire anybody, but they can recommend the Chancellor does, but he doesn't have to. You can google the University of Nebraska regents and see who the eight are.
2. Technically, firing the AD has nothing to do with Riley's status. I believe it is a misnomer to say Eichorst fired Pelini. Eichorst recommended to Perlman Bo be fired, but Perlman fired him. Also, Eichorst didn't hire Riley. He made the recommendation to Perlman. Perlman nominated him to the President, who nominated him to the Board. A national search is only required for the President. Green could fire Riley anytime he wants whether there is no AD, an interim AD, or a new AD who likes Mike. I don't think firing the AD means HCMR is toast for sure, but he needs to win most of the rest of the games and not get blown out to keep his job.
3. Your guess is as good as mine. It probably depends on the rest of the season.