While I do think the rotary air engine itself is a cool device, I would be very surprised it it would be reasonable for this application. The big issues are, how much energy can you store in say a one cubic meter volume? What is the efficiency of converting that stored energy to mechanical energy? What is the expected energy loss from heat flow out of the pressurized tank? Can you recover mechanical energy from the cars motion and re-pressurize the tank? What is the efficiency of that?
These things need to be answered before the air engine can be reasonably compared. Besides it will take energy to compress the air to begin with, which will likely have to come from electric or chemical energy.
One of the feature of using a battery or capacitor is the efficiency of energy storage and conversion, both in terms of unit mass and unit volume.
I don't mean to sound like a jerk, but at first glance, this device doesn't appear to accomplish the desired goals.
Yes, those are exactly the right questions to ask, and compared to batteries, it is less efficient in all the aspects you list. The benefit of this approach over batteries, is that a pressure tank can be rapidly refilled, whereas, today's batteries require at least 4 hours to achieve an 80% recharge, which causes the "range anxiety" that turn car buyers off from buying 100$ BEV's.
I only suggest pursuing this approach, because CA and AZ are pushing for a ZERO emissions vehicle to be sold immediately, and compressed air is the only rapid refill tech readily available, that is also affordable.
Of your list of questions, the only one that I can answer for sure is:
"Can you recover mechanical energy from the cars motion and re-pressurize the tank?" Answer: The existing regenerative braking system would simply recharge the batteries for later use, and not bother converting the electricity into pressurized air.