In fact you could easily see car manufacturers offering such services - something like Zipcar but more convenient as the cars can self redistribute.
I find it hard to believe Uber will ever break even ( not on the day to day costs - recouping all that upfront investment ).
Having said that, the likes of Uber have changed the world - so the money wasn't entirely wasted.
Generally shared cars only make sense when someone drives very little anyway. Cars are expensive and so shared cars are either too expensive to be used often, or very hard to get at. Renting a car for a weekend is about half the cost of a monthly payment on the same car, but if you buy the car you have it the rest of the month (and if you buy a car you have used options or keeping the car after it is paid for to bring the total cost down - you will have to pay maintenance costs but those tend to be much less).
Possibly. However you could also see the Uber type market as reducing the need for people to buy cars - threatening their core business, as well as reducing the number of their potential customers and again shifting the power away from them.
Self driving is the only way uber or a taxi can be a large business. The cost of labor is just too high and so most people are forced to learn to drive.
Even with the above I question if self driving is really worth it. If you own a car and have it parked nearby it is ready to go when you want to leave, and better yet you can store your stuff in it. Combine that with rush hour - most people are traveling at about the same time every day, and some people wanting nicer cars than others and it is hard to see how it can work out. (unless you live in density such that parking is hard - but then transit must be your answer not individual cars since self driving cannot solve traffic)