If electrons are negative, and protons are positive - why do electrons ORBIT? Why don't they just stick?

There is an interesting way to consider this by using gravity, instead of electrical charge to look at it.

Imagine a lot of steel balls held at, say, a foot above a thick steel plate. Lets assume, for this thought experiment, that NO energy is lost, either in the movement of the ball through the air, or when the ball strikes the plate. Gravity will, of course, pull the balls down: vaguely similar to the electron and proton, the ball and plate attract each other.

But as the ball falls, it converts potential energy to kinetic energy, and when it strikes the plate, that kinetic energy is converted into compression of the ball and plate, then quickly back to kinetic energy, and the ball bounces exactly as high as it began. (remember we are imaging a perfectly efficient system. So the ball never "stops" at the plate.

Since the energy of the electron is conserved while the electron is in a single energy state - no energy is gained or lost - it cannot simply stop in a lower energy state, any more than the ball can stop on the plate.

Purists may not like this explanation because it appears to consider electrons as localized entities, but in fact, localized or not, the energy picture is similar.

This is a great question. The answer lies in the fact the the electron at ground state 'holds' a quantified amount of energy and it would not be able to 'hold' that miniumum value if it were closer to the nucleus. I'm not even sure what this means but the answer is in there somewhere.

During my 4 years of undergrad I went through physics waiting to have proven to me that electrons exist. I must say it never happened. No one can say they have seen an electron they can only say they must exist because of many (legitimate) reasons. I always picture 200 years from now scientists looking back at our day and age in science and laughing that we believed such things existed.

They don't orbit. They do stick. The shell they occupy is what is predicted by quantum mechanics as the "ground state", the lowest possible energy state, the closest they can get. That's what a quantum-mechanical object looks like when it "sticks". It occupies a probability distribution instead of a point.

The s-shell of an atom, the probability distribution of the electron in that lowest energy state, is drawn as a sphere but it's not like an egg-shell. It's more of a cloud, and part of that cloud is inside the nucleus. The electron has a significant probability of being inside the nucleus.

And it definitely is not an orbit like a little planet.

What quantum mechanics taught us is that individual particles are not hard little billiard balls. They're blurry.

They do attract each other but the centrifugal force (directed outwards) of electron is just equal to the force of attraction between them. So, the electron is bounded to rotate.

This is analogous to solar system. Earth and sun attract each other due to gravitation but still earth does not collapses into sun because the rotation of earth creates a centrifugal force (directed away from the sun) which is just equal to the gravitational force between sun and earth. The two forces balance each other and the earth remains in its orbit around sun and similarly any artificial satellite remains in its orbit around earth. So is electron around the nucleus.

Your question is quite valid ... in fact this was one of the motivations that later discarded the solar-system like structure of central nucleus and orbiting electrons (they are SUPPOSED TO radiate while orbiting[all accelerated charges radiate] and spiral into the nucleus eventually and STICK there as you say).

We don't picture the electrons as orbiting tiny round balls anymore. We think of them as probability waves. The quantum mechanics tells us that only certain energy levels of electrons are possible in a given atom and the probability waves occupy them.

They do attract each other but the centrifugal force (directed outwards) of electron is just equal to the force of attraction between them. So, the electron is bounded to rotate.

This is analogous to solar system. Earth and sun attract each other due to gravitation but still earth does not collapses into sun because the rotation of earth creates a centrifugal force (directed away from the sun) which is just equal to the gravitational force between sun and earth. The two forces balance each other and the earth remains in its orbit around sun and similarly any artificial satellite remains in its orbit around earth. So is electron around the nucleus.