The way to see what happens is to draw a picture in a frame which is not accelerating. As soon as you let go of the rock, it is no longer being accelerated (except a tiny bit by the wind) so will continue to move with whatever velocity it has. Its velocity is the sum of the velocity you gave it when you threw it and the velocity you yourself had at that time. The result can be odd from your accelerated viewpoint... but the rock is traveling in a straight line to an observer that is not spinning on the ride. So, for example it is possible to toss the rock directly toward the center of the ride and then catch it as it "falls" back. However, if you were above the ride and watching (but not spinning on it), you'd see the rock tossed at an angle (the velocity of the toss plus the sideways velocity you had when you tossed it) and then it moves in a straight line to where you catch it. (You followed a curved path to get there-- which is why you felt the pressure from the walls of the ride). If you did not catch the rock and it missed the walls of the ride, it would simply continue in a straight line away from the ride.

Yes -- if you are truly falling, the rock will float. However, do not try this -- if you let go too early, (while you are still accelerating, the rock will fall rapidly, and if the cage you are in is not exactly free falling the rock can be left a long ways in the air -- to later fall on you when you land... This is precisely the same think that happens to objects in orbit around the earth -- they too are 'falling' it is just that their lateral velocity is so high that they miss the earth.

Those are excellent questions and I am excited that you are thinking about physics while you are at amusement parks. The question about the gravitron involves a concept called centripetal force or centripetal acceleration. The concept is difficult to describe so if my explanation is confused you can easily find a number of informative web-sites by typing centripetal force into your favorite browser.

The gravitron is basically the same as taking an object on the end of a string and spinning it above your head, with the main difference being that you are supported from behind by a wall rather than from the center by a string. Luckily, this makes no difference in how the concepts are applied. If you were to see the spinning object from above it is clear that the path of the ball is a circle. At any instant in time the velocity of the ball is a straight line tangent to this circle. This can be visualized by cutting the string as the object is spinning above your head. It flies off in a straight line from the point where it was when the line was cut. If you are in the gravitron and throw the ball up, so it leaves the gravitron, it would do exactly the same thing as cutting the string on a ball above your head. I realize this is slightly boring, the ball goes straight out of the gravitron.

The more interesting case is if you were to throw the ball towards the center of the gravitron. In this case the ball still moves in a straight line but since the people inside of the gravitron are still under the centripetal acceleration the ball will appear to fly off in one direction or the other.

Depending on the size of the gravitron and how hard you throw the ball towards the center you could hit someone to the left of you, someone to the right of you, or even yourself! Be careful!

Whenever you throw something, it follows the standard parabolic path from the instant it leaves your hand (neglecting air resistance, of course).

So, all you have to do is consider what that path looks like relative to your own motion. To do this, it helps to consider what your motion and the ball's motion look like relative to the earth.

Inside the Gravitron, you toss the ball gently upwards. You are moving sideways with the ball, so you don't see the ball move sideways. Now suppose the Gravitron is transparent. Someone from the outside would see the ball moving like a pitched baseball- very fast sideways motion, plus a little upwards motion as it leaves your hand. From the outside, the ball is moving so fast it traces a nearly straight line, and its path over the ground is perfectly straight, as usual. Since the Gravitron is a closed space, it is obvious that any nearly straight path must soon hit the side of the Gravitron. Since you tossed the ball up, it will arc over the top of your head and hit the wall behind you. You tossed it up, but instead of falling back into your hand, the ball went backwards!

This just an illusion caused by your motion: While the ball followed a straight line over the ground, the wall of the spinning Gravitron pushed you over that line. The ball's motion appears mutated, but that's only because of your mutant motion.

Another trick to try in the Gravitron is to throw the ball towards the center. It veers away from the center and strikes the wall somewhere to your left if the gravitron spins clockwise, or to your right if the spin is counterclockwise. The ball appears to curve, but again this is an illusion caused by your motion. In a transparent Gravitron, the outside crowd would still see the ball fly as normal, with a straight track over the ground. The reason it curved to miss the center is because you didn't actually throw it at the center, even though you thought you did! You miss for the same reason you miss if you throw a ball straight out a moving car's window at the instant when your target is beside you: The ball has your own sideways motion, in addition to the motion you give it.

In the case of the Gravitron your sideways motion is invisible, but the ball still has it when it leaves your hand. Next time you're at the carnival, try going in the Gravitron with a bunch of friends. You can throw pieces of candy upwards, at the center, and at each other to see what happens. Don't bring a ball, because it will probably get pinned to the wall above your heads or in some other place where you can't get it until the Gravitron stops.

On Superman, to get the ball to hover in front of you, you simply let go of it as soon as you feel totally weightless. In this case anyone watching from the ground would see you and the ball as two objects both tossed upward, side by side, and moving together. Wind may spoil the effect, and if you throw the ball or release it before you feel totally weightless, its strange motion will not be as obvious because it will move away from you.

If you were in a version of Superman that was enclosed like the Gravitron, there would be no way to tell whether you were in the ride or in a UFO in weightless space. Released balls would hover, you could drift around and slurp from big floating globs of water, etc. Without peeking outside in some way, there would be no possible experiment you could do to distinguish between falling vs. a total lack of gravity. In fact, at 300 miles up (the orbit of the space station), gravity is at 85% strength- but you still see the astronauts doing all their "weightless" tricks! That's because the station is actually "falling" in a circle!

The station orbits at the incredible speed of 5 miles per second. A dropped object will drop about 15 feet in the first second of its fall, 5 miles away in all directions, the ground curves down by 15 feet due to the round Earth.

Normally, things thrown sideways still fall and hit the ground. However, if something shoots sideways at 5 miles per second, the earth will curve away as much as the thing falls. That is an orbit- a fall that never stops. So, just like in the Superman ride, the astronauts have the illusion of weightlessness even as they fall in nearly full-strength gravity.

I wouldn't ever go on a spinning ride because I like my stomach right where it is (interesting that you say "throw something up in the air") but here is what I think would happen:

If you are holding an object, like say a penny, and let go it would be similar to the penny being tied to the end of a string that you were swinging around in a circle and let go. Thus the penny should start moving straight in the direction you were going when you let go plus it should start to fall due to gravity. So the penny should hit the wall pretty quickly. At that point, I'm not sure if it would stick due to friction or bounce and slide down the wall to the floor.

Many years ago, I went on a ride call "Free Fall" at Magic Mountain and tried your experiment (probably frowned upon by the staff of the park). The idea of this ride was to bring you up to the top of a tower and then drop you while sitting in a box/car sort of vehicle. So it was easy to place a penny on your knee and watch it as you fall. Sure enough, the penny "floated" for the short time we were in free fall. Why do you think this happened?