It doesn't appear to me that you have enough headroom for a bathroom. You could fit in a half-bath with just a water closet and a lavatory if the person using the WC is short of doesn't mind sitting down to do #1, but I'm not seeing a lot of standing headroom in there.
As to the other questions:
Plumbing seems doable. Since the garage is below, i can just run exposed plumbing and wiring through the floor from the garage.
No, you can't. Wiring, yes, with all holes caulked with firestop caulk. Any garage that's beneath living space (even a "future" bonus room) has to be separated from the living space by a continuous layer of 5/8" firecode sheetrock. Plastic piping (which is what I'm sure you would use, since that's what everyone uses today) doesn't hold up when exposed to a fire, and once it melts it creates an opening for the fire to reach the room above. You can't leave the piping or the penetrations through the floor exposed to the garage.
I'm just not sure what to do about the ceiling in this space. Can I just put in fiberglass insulation and screw drywall directly to those rafters, or do I have to continue an air path between the eaves and attic so the "real" attic can breath?
How deep are the rafters and how much insulation does the code require where you live? You DO need a continuous path for air flow, but it only needs to be about 1-1/2 or 2 inches deep. It's a code requirement, but one that's there for a good reason. No air circulation ==> condensation ==> rot. It's a common problem around here, especially in houses with cathedral ceilings.
Also, if I want to get a permit for this work, do I have the city inspector come out preemptively and approve it?
"Preemptively"?
The permit is issued based on a written application that describes the work to be done, and plans that show the work to be done. Inspections take place in stages as the work progresses. It's already enclosed, so my guess is there would be a "rough-in" inspection when the rough electric and plumbing are in place, an insulation inspection when the new insulation hs been Installed (it cannot be installed before the rough-in inspection has been performed), and a final inspection when the work has been completed.
Lastly, it's unusual to see a narrow attic like that with a floor. If that strip of floor is to provide access to something like an air handling unit behind the position from where you took the photo, the access is required for maintenance and you can't block it off by building a bathroom ... or even a closet, unless the closet also has an opening for access to the catwalk.
