Why are planetary orbits in near perfect equilibrium rather than falling into the sun or escaping the sun's pull?

The velocity vector can change wildly and still result in a valid orbit--it's just that the orbit will no longer be circular. For example, if you slow down the Earth, it would transition to an elliptical orbit with its aphelion (furthest point in the orbit) at the current Earth distance (1 AU). The perihelion (near-point of the orbit) would be closer to the sun. The more you slow it down, the closer to the sun it would get. But the orbit would be stable (it would NOT spiral in unless you kept on slowing it down). At some point, though, if you slow down the Earth enough, its perihelion would be inside the Sun's surface. That would not be a stable orbit. Conversely, if you speed up the Earth, it would transition to an elliptical orbit with an aphelion beyond its current orbit. Comets are orbiting bodies with perihelions inside Earth's orbit and an aphelions out beyond Pluto. They are still stable orbits. If you speed up the Earth it would start to follow those kinds of orbits. It's true, of course, that at some point you could speed up the Earth enough to reach escape velocity. But the range of valid speeds is still very high. But this doesn't totally answer your question because all the planets have nearly circular orbits (only slightly elliptical); and you have to have a very precise speed to have a circular orbit. So how did the planets get the exact speed for a circular orbit? The https://en.wikipedia.org/wiki/Nice_model (after the French city) is one of the best theories for planetary formation. It assumes that after the giant planets formed, they fell into resonance with each other, but disrupted the other planets and caused many remaining asteroids to scatter (some of them later crashed into Earth). At this point, the orbits of all the planets were highly elliptical (as you would expect). But there was still a little bit of gas left over from the original cloud that formed the Solar System. This cloud was not evenly dense--instead, it was denser close to the Sun and thinner farther out. That's not too surprising, right? Now imagine a planet like the proto-Earth with an elliptical orbit passing through this cloud. At perihelion, when the proto-Earth got close to the Sun, it would have to pass through a dense cloud of gas. This would cause proto-Earth to slow down! When you slow down an orbiting body at its perihelion, it robs it of energy and makes the aphelion closer to the Sun. Conversely, when the proto-Earth reached aphelion (furthest from the Sun) the gas was much thinner, so it didn't slow down very much.[1] The net effect was that proto-Earth's aphelion kept getting closer to its perihelion. The closer together the two got, the more circular the orbit got. Eventually, the gas cloud dissipated and the planets were left in stable, circular orbits. [1] Of course, the gas did slow down proto-Earth even at aphelion. The result would bring perihelion closer to the sun, but this effect was much less than the effect at perihelion (because gas is thinner out at aphelion). Nevertheless, this caused proto-Earth's orbit to slowly spiral in. Once the gas dissipated, the orbit stopped changing.

If you have a single star and a single planet then things are actually fairly simple: if the total energy of the planet is negative, and the orbit is not so elliptical that it actually passes through the star, then you have a stable orbit that can easily remain essentially unchanged for billions of years. (For the purposes, the total energy consists of kinetic energy and gravitational potential energy, and the potential energy is normalised to 0 at infinite distance, and is negative for closer distances.) Since an isolated planet has almost no opportunity to gain or get rid of either kinetic energy gain or or angular momentum, it has little choice but to keep going round in some form of ellipse (the simplest case of which is of course a circle). Where it gets complicated is when you have multiple planets or other smaller objects orbiting a star. Since they're all going round in different orbits at different periods and potentially different orbital planes and ellipticities, they have many opportunities to interact with each other and perturb each other's orbits. So over the long term, many are going to collide and fuse, and the rest are going to be ejected from the system entirely. The most perturbation-resistant combination is a small number of planets going round in the direction in the same plane. In particular, because of the phenomenon of http://en.wikipedia.org/wiki/Orbital_resonance , any planet will tend to sweep its own orbit of competitors, as well as any orbits with simple whole-number ratios of periods.

Orbits are pretty stable. You can characterize an orbit by energy and angular momentum.  As long as those don't change much, and it's relatively hard for them to change much, the orbit won't do anything crazy.

To summarize previous answers, those planetoids that were in unstable orbits have long since gone away. Orbits that were stable remain stable because there is negligible air resistance/drag in space. Once in orbit, if a planet gains or loses energy due to collisions, it could be shifted to a higher or lower orbit, or one that is more or less eccentric (like an ellipse rather than a circle), but it will end up being in a different stable orbit than before. In order to become unstable a planet would either have to lose all of its orbital velocity through collisions, or gain escape velocity. Any force strong enough to do either would likely tear the planet apart first.

Planetary orbits are stable because of the inverse-square properties of gravity.   if it was linear or cubic orbits would be unstable and very quickly diverge.

I thought about this question a lot and it seems like its just as simple as "natural selection of planets" or the Stuff (Gas) that flew around the sun before forming the planet's that didn't got sucked up by the sun and had time to accumulate into big clouds that build up gravity thus absorbing more stuff and building a planet For experimentation and getting a better intuition of how the interaction of orbiting planets is, i recommend this simulator https://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.html If you have an Android cell phone you should try this free app https://play.google.com/store/apps/developer?id=Orbit+Simulator

https://en.wikipedia.org/wiki/Dynamical_friction tends to round out orbits as long as there are a bunch of relatively small objects to encounter. At the top of an elliptical orbit, you are moving more slowly than an object in a circular orbit at that distance, or than the average of objects there in various orbits. Encounters will, on average, take away some of their mean velocity and give it to you. Conversely, at the bottom of your elliptical orbit, you are moving faster then the speed of a circular orbit at that distance, and encounters will on average reduce your speed.

Great question.  Every question I answer I need to rehash the idiocy of  the totalitarian science that was spear-headed in the early twentieth century. Let me rephrase your question.  Why is gravity so forgiving and UNCLUMPY.   Let us recognise that where two major bodies are concerned gravity works in such a way as not to be the CLUMPY force that the formulae would suggest.  Where we have two small bodies the force of gravity is weak but clumpy.  Where we have a large natural body, and small artificial body,  gravity is STILL clumpy.  For example, Phobos, an artificial body, hollowed out,  conforms to the clumpy formulae, and on its own will crash to Mars within 11 million years, and I would suggest sooner. The mystery ends when you realise a number of things.  Gravity is a mechanism.  Its not a set of formulae.  And the formulae don't work. They appear to work within a certain range.  Outside that range they get less and less viable as an approximation of what is likely to go on ...... The next thing to realise is that while in a terrestrial sense, the various CONSERVATION laws, are pretty well-founded .... they must be rejected as they are a rebellion against logic.  Nothing is conserved.  Yes perhaps locally we have conservation.  But in the more wide sense nothing is conserved or nothing would exist.  There is simply no getting around this logic. Reality is made more plausible by a situation where gravity is CLUMPY and UNFORGIVING where small amounts of matter are concerned.  But reality is made more plausible so to speak if big amounts of matter under gravity are unclumpy and forgiving as to their orbits.  We can answer this story by looking for likely areas where we get more MASS, more ENERGY ..... more momentum and more angular momentum. You see for gravity not to be clumpy with two substantial bodies, no way can it just conform to a long-dead, unfulfilled homosexual alchemists formulae that he purloined from a better scientist.   It just can't work that way, even if we were to sacrifice a bull to Newton, every month for a thousand years. If matter was unclumpy with the small stuff then we could not have reality because there would be endless orbits of unsubstantial material.   But should gravity be clumpy, as the formulae suggest, with two large objects, then there is no capacity for habitats for biological life.   I mention this partly in passing because while I have good prejudices towards theists thats not the point here. We see in the unclumpy and forgiving nature of the orbits of two large bodies,  as the best crime scene to reverse the illogical conservation doctrines in all their forms. I will leave it at that, and I expect the usual suspects, to run a spam campaign against what I'm saying.  And I hope for intelligent questions.