If there will be heat generated in the mechanical room, I advise insulating the wall between that and your root cellar.
I am not sure what to say on the other walls. I am an architect in the hot, mostly-basementless south. Down here I would probably opt for placing the root cellar in a corner, against two adjacent perimeter walls in order to maximize earth contact. I don't know what the temperature range your cold-climate unconditioned basement might experience, but I suspect it must stay cool all year since you located the root cellar away from perimeter walls.
Since the floor of your basement is below your frost line, and your root cellar is located away from the perimeter, I see no reason for floor insulation. In fact, it would seem that an uninsulated floor would provide your only connection to the wonderful cool thermal mass of the earth. Definitely insulate between the ceiling of your root cellar and the living space above.
I have limited experience with basement root cellars, so maybe someone with more direct experience will chime in. (I'm jealous of your root cellar. Maybe someday I will have one.) Good luck and enjoy!
