pool physics
Whew, so formal. :o
To start with I think we are talking about something like a Beach Ball in an indoor swimming pool. If this is wrong, please advise. This problem is more of a materials problem than a Physics problem, so please be specific.
Background.
Beach Balls are a lot less elastic than a lot of people assume. Their thermodynamic efficiency in reacting to an impulse can be less than 66%. There are many factors that affect this. Air pressure, internal pressure, the age of the ball (they loose elasticity with age), temperature, and humidity to name a few. Add to this the fact that the weight of the ball can be substantially different from it's mass due to its displacement in the air and you wind up with a problem that requires a substantial amount of experimentation, measurement, and math to accurately describe. Add to this the fact that the results will probably only apply to a specific ball under extreemly narrow conditions and you wind up with a problem that is best answered qualitatively rather than quantitatively.
Here are my qualitative answers in order.
* It starts rolling as soon as it starts sliding. The viscosity of air and water are substantially different. As soon the ball starts moving this difference will start the ball turning. It never really slides then rolls, it starts rolling as soon as it starts sliding.
* There is a VERY complex relationship between the circular momentum of the skin of the Beach Ball and the circular momentum of the air inside. They would both be transferring energy and momentum constantly. This makes the results kind of inconsistent as far as predicting an exact trajectory the way you can something like a baseball or even a rock. That being said, Bernoulli's Equation answers this. (See Bellow) The side of the ball that is going against the flow of air has a higher pressure acting on it than the side that as going with the air flow. This makes the trajectory curve toward the side that is coming back to the thrower.
* This can be described using a vector quantity that describes rotation on three axis. This is called Curl in Vector Calculus. Vector Calculus is usually at least touched on in most College level Calculus Book.
* Bernoulli's Equation answers this. K=P + (p*v^2)/2 +p*h where K is a constant, P is pressure, p is density, v is velocity, and h is height. If we assume that h does not change, we get P = K  (p*v^2)/2, or, the Pressure acting on the wall of an object moving through a fluid is proportional to a constant minus one half the quantity of the density of the fluid times its velocity squared. Water is 1000 times as dense as air, so for any given velocity the pressure acting on the ball is 1000 times greater on the parts in water than the parts that are in contact with the air. This actually makes the ball tend to sink as it is accelerated. As soon as you get enough force to make the ball clear the water, this suction from the water goes away and the ball, which is slightly compressed because of the forces applied to it, springs up.
* In reality, viscosity is the main force at play. The wetted area, the drag coefficient of the material that the ball is made from, and Reynolds numbers of the fluids as they pass over the ball are the main factors for computing losses and forces. Objects moving through newtonian fluids such as low velocity air and water really don't have a static frictional force to overcome the way something like a brick sliding across a concrete surface would. This is why a single man in a dead calm can easily push and pull a 40 ton boat around.
I hope this helps :)
