I have quite a bit of knowledge of roof venting in the northeast where ice dams are more of a concern, but you still need to vent properly in your climate.
By far the easiest solution would be to install ceiling/floor joists and insulate between them, then sheetrock. Venting would not be a major concern, but I would add gable vents to pull in outside air. If you want to do the cathedral ceiling, you need to install the Styrofoam baffle (called proper vent around here)the entire length of the bay and add soffit vents and a ridge vent. If you have any hips and valleys that will not be vented by the soffit and ridge vents, drill 1 1/4" holes along the rafter so it vents into the adjoining bay.
If mice can get in your garage, then they will get into insulated rafters. Insulation doesn't bother them, so seal all possible entry points and don't forget the seal under your overhead door where they commonly get in in an otherwise new sealed building. Good luck!