You've asked three very distinct questions here. I'll try to answer them all.
Could this scenario even happen with Earth's gravity and atmosphere?
I doubt it. For one thing, most of the ring material would burn up in the atmosphere, much as meteors and deorbiting satellites do, never reaching the ground. Second, even if you manage to handwave the atmosphere problem, there's a reason why there are no mountains on Earth bigger than the Himalayas. If Earth did somehow have a mountain range twice as tall as Everest, the weight of the mountains would overcome the shear strength of the rock they're made of, and they'd pretty quickly (in geologic terms) slump down to something closer in size to the Himalayas. I don't know how long this would take, but the slumping would probably go faster than the mountains would build up, so the ridge would never get much taller than the Himalayas are now.
How long could this ridge last in Earth-like conditions before the forces of wind and erosion eat it away?
Probably about as long as any other mountain range... however long that is. Quick Internet suggests somewhere in the neighborhood of several hundred million years after the ring system stops building the ridge up. Maybe. This could be a question for another StackExchange.
Are there any unique changes that would occur to Earth because of having essentially a huge mountain range circling the Equator?
For one thing, the ocean currents would be severely disrupted. You wouldn't get seawater circulating around all the world, simply because there's a huge mountain range around the equator that water cannot flow through. I have no idea what impact this would have on the global climate, although it's possible that one hemisphere could be warmer or cooler than the other to an extent that is not possible without an equatorial ridge. This could make the entire Earth more prone to having ice ages.
If the Earth's axial tilt and orbital eccentricity line up such that the Northern hemisphere has harsher winters than it does now, snow and ice will build up in Europe, Asia, and North America, reflecting sunlight back out into space and cooling the planet further. This leads to more snow ice in the Northern winters, creating a positive feedback loop that culminates in an ice age. When the Earth's axial tilt and orbital eccentricity lining up to give the landmasses of the Northern hemisphere more severe winters (and summers), they also line up to give the oceans of the Southern hemisphere a more temperate climate. The oceans may be able to moderate this effect by exchanging water between the hemispheres; but if there's a tall equatorial ridge, that could not happen.
Also, it would become much more difficult for air to pass from one hemisphere to the other, but I don't think this would have as great an effect on the global climate as disrupting the ocean currents would, simply because not a whole lot of air crosses the equator anyway. All of the major convection cells lie in one hemisphere or the other.
The ridge would also have a major impact on life on Earth during the (relatively short) time that it would exist. Very little in the way of plants or animals would be able to cross the ridge, so life in each hemisphere would almost certainly follow very different paths. 100 million years ago, dinosaurs roamed the Earth. 200 million years ago, the Triassic gave way to the Jurassic. 400 million years ago, the earliest ancestors of all modern land vertebrates (amphibians, reptiles, mammals, birds, etc.) hadn't even come out of the oceans.
However, do keep in mind that if you want to keep the two hemispheres completely separate, so they can follow utterly different evolutionary paths, you'll have to make sure the ridge doesn't get broken up as the tectonic plates beneath it move around. I don't know how you might go about ensuring that. If you just drop more material on the equator continuously, then you'll have less of a clean ridge and more of a messy system of massive ridges and plateaus. Plate tectonics will carry what exists of the ridge away from the equator much more quickly than it will erode away. Also, you'll get gaps where plates are moving apart, such as in the middle of the Atlantic, where living things will be able to pass through.
