If it's not Uber, someone else offering a similar service will eventually allow people to submit their own self-driving car to a driving pool when they're not personally using it.

Or people will buy a car and submit it to Uber's pool for some return, as an investment. Minimises Uber's costs which will mean they can expand aggressively.

No because the cars will be the electrical storage for most of the non baseline load in a city. So cars will be part private property and part public transport - fleets of mini vans driving around and nipping off to the charging station at peak demand times.

No, all of that could still be outsourced to a 3rd party.

The gig economy will only die when humans price-gouge each other so immensely that robots are the only viable option - even though they can carry a higher entry cost.