Just as an exercise, because I've thought about it off and on for years, and have never quite wrapped my head around it, I would love some input on designing a "flat world" campaign setting.
Done a lot of thought on worldbuilding. (So much so I've
written a book on it.)
So happy to give some input.
1) the world has an ocean in the center, a ring of mountains on the exterior, and rivers that run down from the mountains to the ocean.
Sounds good so far, and solves the problem of water running over the edge. A "saucer" shaped world.
2) There is no sun, and no moon(s). Day is simply "when the sky gets light" and Night is the opposite. There are times of "nightglow" but there is no actual moon (unless we invent a way to have one).
This is where I start to question you. Gently.
As a rule, the more funky changes you make to the world, the more careful you have to become, because this can cause odd ripple effects.
Like how to werewolves work without a moon? Are there tides? Why is the spell
moonbeam? How do magic items that recharge "daily each dawn" work when there is no literal dawn?
Would there be seasons if there is no sun? Do days still get warmer and hotter? Where is the source of said warmth?
Without a direct source of light, are there shadows? Which way do they fall?
This all seems trivial until you have a way of the shadow monk in a light forest asking which way the shadow of the trees are falling so they know where they can teleport.
3) burrowing through (or climbing over) the mountains in the outer ring would only end up "coming out the other side"; the dwarves probably make a lot of money creating travel short-cuts this way.
Do you mean they just emerge on the far side like the edges look around and connect? Like an older RPG video game map?
Questions: Seasons? What's "up there?" and "down below"?
Up in the sky? Down underground?
Or are you asking about the top side and bottom side of the saucer?
There could easily be stuff on both sides of the saucer. You could do traditional fantasy on one side, and non-traditional on the other.
Or have opposing factions with different races, like Warcraft, with an orc nation below. So elves and humans are only on the dorsal side while gnomes and dragonborn hail from the ventral.
What does the source of a river look like?
Like the real world.
Clouds are pushed up by air currents hitting the mountains, and cool in the upper atmosphere. Rain comes down and makes small brooks and streams that merge into larger rivers.
Are there other planes? Is there an underdark?
No reason there couldn't be other planes.
If the disc is thick enough, there could easily be an underdark.
The deepest real world mine is less than 4km thick, so a 10km wide crust would have plenty of room. (The highest building in the modern world is still less than a kilometer tall, so 10km gives you more than enough room for a dozen medieval cities stacked atop each other.)
Is there a "backside" to the world disc? Is there really even a "disc" if there's a wraparound effect?
See above for "backside" thoughts. No reason there couldn't be.
Is it a disc? Well, it's almost more the inside of a sphere. (See
Hollow World). But since fantasy people might not be able to map everything, the disc/sphere question is effectively an armchair philosophical question for people in-world.
It doesn't matter in play for the most part and could just be a "mystery".
