That's a great question. We have what's called a vestibular system in our inner ear that helps our bodies know if we're standing, lying down, moving, etc. It's important for our brain to know these things so that we can be coordinated and balanced (imagine trying to walk properly if you couldn't tell which direction was up or down!). Dizziness should definitely affect hand-eye coordination because your body wouldn't be able to correctly orient itself to conduct synchronized motion. A lot of what we perceive about the world depends on how our brain interprets it; if your brain thinks you're spinning, and has you seeing the world as if it were spinning, I don't think you'd be able to, for example, successfully shoot a basketball in a hoop.

Here's some more information specifically on why we get dizzy. There are canals in our ear that allow us to sense motion, and these canals are filled with endolymph (fluid) and hair cells (cells that send sensory information to the brain). When you move your head, the endolymph moves the hair cells, which then send information to the brain to interpret what kind of movement that is (e.g. "Oh, the hair cells are moving at this angle, so this must mean the head is turned to the right!").

When you spin, the endolymph moves in the same direction of your spinning; when you stop spinning, the endolymph continues to move, stimulating hair cells to send signals to the brain. The brain takes these signals and interprets them as the head is still spinning, even though you have stopped moving. This is why we get dizzy! When the endolymph finally settles and sits still, the brain doesn't receive any more spinning signals so you don't feel dizzy anymore.

I'm not sure if longer spinning would make you feel dizzy for a longer period of time; perhaps spinning faster would cause the dizziness to last longer because a slower motion of endolymph will settle back to normal faster than the speedy endolymph. You should try this out as a mini experiment!

That's a good question. Dizziness is caused by your vestibular system (which senses motion) being out of time with your true motion. You sense motion using 3 semicircular canals in your ears that are at right angles to each other. They contain a fluid (endolymph) and hair-like sensory nerve cells. When your head is moving (e.g. by spinning), the endolymph fluid moves at a slower rate (due to inertia), which stimulates those sensory nerves, which send signals to the brain that you are moving in a certain direction. When you stop suddenly, your brain still thinks you re spinning because the fluid is still moving... and you feel dizzy until the fluid stops moving and your brain has the signal that you are still. This would affect hand eye coordination until the dizziness passes, because your mind thinks the body is in motion. So the more the fluid is moving (the faster the spin) at the time you stop, the longer it will take for the dizziness to pass.