In the context of building the team, yes. He does a lot more than that behind the scenes, sure.
Aside from the HC, correct. I'm sure he's involved with some other staffing decisions i.e. the scouting department, training, etc but I'm speaking more towards the coaching staff and player personnel.
The buck stops with him in regards to the terms of trades, compensation and such, but the HC is the primary driver of proposing trade ideas.