Ambulances do a lot more than transport patients. It's already the case that you can usually get someone to the hospital faster than waiting for an ambulance. Typically if you're using an ambulance you're going to need someone to provide medical assistance on location and/or help load you onto a stretcher.
That depends on how far out you're projecting. In a hypothetical all-self-driving future, a car with a sick occupant could broadcast its emergency mode to the rest of traffic to let it through.
Well, the car can't go through red lights or overtake in certain locations to get to the hospital more quickly. That could be an easily ignored issue in some cases (if traffic is light or if it's easy to get to the hospital from your current location), but good luck if there's a traffic jam or the car needs to ignore certain laws to save your life.