Krdytkyu

I've looked for an answer for this, but the only information I could find dealt with the ring itself. All I want to know is what would change on the Earth if all problems were solved and the ring were there to stay.

In this particular world, the geography has changed due to a total destruction of Earth's surface, with humanity evacuating to the ring. A secondary, magical humanoid race has restored the planet to nearly it's original state.

What I want to know is if any biological/geographical adjustments would need to be made due to the ring's presence, and what the climate zones would be.

Thanks to the shell theorem, and assuming your engineers did their jobs, the only affect is on how light gets to the surface

The Shell Theorem states that the gravity caused by a uniform mass above you (e.g., a ring or a dyson sphere) has no impact on you, the person inside the uniform mass. Only mass "below your feet" has an impact. Think of it this way. If a straight tunnel magically existed from Earth's surfact to its center, then as you descended the tunnel, only the mass still between you and the center affects you. Thus, the gravity affecting you gets less and less until it's finally zero at the center. (Which, BTW, argues quite well for something other than a liquid or solid core at the very center. Probably a really hot plasma having fun in zero-G.)

Now, this assumes your engineers did their job designing the ring. If you read Larry Niven's Ringworld Engineers you'll discover (because he was famously informed about it by a bunch of MIT students in the 70s) that rings are inherently unstable. You need engines to keep the ring in place. One assumes the engineers were bright enough to place the engines and design the size of the ring such that the engines don't fry the Earth.

That leaves sunlight, direct and reflected

Therefore, the only issue is the shadow cast by the ring during the day and the reflection of light off the ring during the night. However, for this to have a significant effect (other than on romance... I can see entire jewelry commercials being inspired by this ring) the ring would be required to have an enormous width. It would need to be, at a guess, thousands of miles wide just to create a discernable line of shadow on the Earth. This all depends on its radius compared to Earth's (for a fun exercise, see this question that deals with this from another point of view).

Conclusion

In reality, the total effect of the ring on the Earth is negligble if not outright ignorable. It might create a cool sky effect (depending on its size), but that's it. This isn't surprising as natural planetary rings (e.g., Saturn's rings) don't have an measurable impact on the planet (that I'm aware of... Fair point, I'm not perfect).

If the ring lies perpendicular the the planet's axis and of a certain width, it could cast a permanent shadow onto the earth, making photosynthesis and therefore plant life impossible under this shadow.
The shadow could also cause other effects in non-permanent regions in the form of a short "artificial night", which may confuse fauna and flora, cause the temperature of the region to drop and mess with the weather (air temperature plays a role in the climate's behavior).
Should the ring have enough mass, it may even affect the tides, further causing interesting effects together with the gravity of the moon.
Again, if the ring is perpendicular to the planet's axis and has enough mass it may encourage higher trees or mountains due to the effects on gravity.
Should the ring contain large, lens-like segments (I dunno, maybe for observation or heat harvesting or technobabble), it may also have an effect on the sunlight reaching earth (think giant lasers when it lines up with the sun).

In the end, the bigger the ring, the stronger the effects. Look to the moon for inspiration. Hope this helps.

It would slow down the rotation of the Earth. I'm assuming the ring is around a line connecting the sun and the Earth to prevent the ring from blocking light from getting to the Earths surface. The consequence of this is that the Earth would slowly start to drag the ring to rotate along with itself. The reason for this is complicated but we see it happening with our Moon. The Moon creates a swelling of water on the Earths surface directly beneath it (the tide) the spinning of the Earth, which is faster than the orbit of the moon, pulls that tide in front of the moon. The mass of water then imparts a small force causing the moon to accelerate and the Earth's spin to decelerate. (a quick internet search pulls up this if you want a reference https://www.bbc.com/news/science-environment-12311119). Your ring would create the same effect (this also means it would require rockets on it to prevent its spinning).

This could potentially be fixed by having the ring rotate with the Earth and eat the cost of having a solar eclipse every day.