The biggest question for the administration to answer, in my mind: Do they support the hockey program, or the leadership thereof? Sometimes you can get the greatest guy in the world to be the head coach, terrific "face of the University", really involved with the community, great on outreach, get good recruits to come in, but be terrible behind the bench with game management, and it only takes a couple down recruit years to expose that. If the admins have fallen in love with the coach rather than the program, they could be looking past that and be a source of trouble. How tough is the administration on his contract? When does it expire, or worse, was he "tenured" like a professor? I don't see any info quickly looking online.
First question to ask: What is the vision you want on the ice to be competitive, and is the coach satisfying that?