You can't switch balls in the middle of a round. The game will continue using balls of the type you started with until you run out of them (either by using them up or losing them.) The game will then give you the option to buy more of that type when you run out. If you decide not to buy any, the game sticks you with the Starter ball for the rest of the round.
How much you can play on a sleeve depends on the Durability rating of the ball. I don't know if anyone has counted the number of hits you get with a particular ball. There's just that little graph that gives you a rough idea of how much life is left in a ball while you're playing, and you get the message when there's one hit left.