I had to go look up what each region is roughly expected to be like. Noting that the plane is supposed to be finite, you homed in towards the problem with that hunk of rock, as far as I know it's impossible for there to be a cold icy North like there's supposed to be and a forest in the East. I can come up with complex scenarios to physically get either/or but not both. You are thus required to invoke pole-based magic.
As we are now invoking magic for weather you can get whatever you want from the system. As we have magic weather we might as well choose between uniform gravity and gravity-as-usual where you would feel more of a pull towards the center of the plane as you ventured towards the edges. With arbitrary magic atmosphere I see no reason not to magically mix breathable air up to any height. In fact I can't see any real reason for "space" to actually ever start. Certainly at some point the atmosphere wouldn't be bound by gravity but that's no reason for the "beyond" to be anything other that air.
Actually I just revised one of my scenarios and there is The One where you could achieve the necessary conditions realistically. You need a practically solid mountain range around the South and one separating the East and West on the topside of the plane (your center mountain) to set up Orographic rain for the East's forests.
You would need a sun that orbits extremely low over the Southern end so that it causes the central mountain to cast a shadow over the North nearly all the time. You would also need the bottom side to be entirely flat ocean so the rain shadow effects have time to be canceled out and evaporation to occur for for the North.
Wind would continuously blow West and weather pattern would be unchanging from day to day.
Due to the low Southern sun and the constant Western wind it would become impossible to get lost pretty much everywhere. Your crazier weather all but disappears except in the north where ice crystals and crazy mixing from the dual rain shadows (your finite edge of the plain acting like mountains) cause intense storms.
So "How could normal humans deal with the issue of a giant mountain causing world-ranging problems, without resorting to prayer or summoning supernatural entities?"
In crazy pole-based magic land they observe any pattern in the magic if they're there and exploit them, or stay as far away from each pole as possible if their particular effects are dangerous.
In The One land you have guaranteed constant weather so you find a place you like and stay there.
It was asked at what angle the sun should be over the South so I figured I'd flesh it out:
It's worth noting you won't have seasons with this setup as calculated. You could have a "wobbly coin" for your plane relative to the sun to pull them off but you'd have to tune everything carefully to get season while still keeping the ice in the North and the desert in the South.
Note that ALL references to distance are in regards to the Northern Edge.
Alright, so the major cause for temperature changes is insolation. Ice has natural reflectance, so our target is enough shadow cover to get to 0 °C and we'll assume a feedback loop takes over from there and drags it lower and stabilizes it.
$$\text{Temperature} = \sin (\text{SunAngle}) \times \text{SolarConstant} \times \text{PercentDaylight}$$
I'm not going to use the typical definition of solar constant here, mostly because we don't care, it's a raw solar heating number that gives a target base temperature.
Now the mountain is 600 miles tall and based on looking on some Exalted maps I'm going to guess ~30,000 miles average diameter for the plane.
Using trig you need that shadow to cast from 600 on one edge of the triangle to 15,000 on the other so we can cover the widest part of the north
$$\tan ^{-1} \left( \frac{600}{15,000} \right) = 2.3°$$
(thats maximum sun height in the sky from the far edge of the North)
This should also maintain the amount of shadow by sweeping equally across the North. Assuming 24 hr days we have 12 hrs of shadow time roughly equally distributed across the north. So subtracting, the North receives 11/24 daylight percent.
In order to have the shadow sweep across the entire North our sun's orbit radius must be equal to the radius of the plane. This is because as we head towards infinity the shadow from the mountain will actually spread across the entire top half of the map. And as we head towards a radius of zero it will tend toward a straight shadow behind the mountain...
Using our plane radius of 15,000 miles:
$$\text{Distance} = \frac{15,000}{\tan(\theta_1)}$$
The middle needs to be 25 °C and the north 0 °C.
Using kelvin to avoid 0 °C in math:
$$273 = \sin(\theta_1) \times C \times \left( \frac{11}{24} \right)$$
$$298 = \sin(\theta_2) \times C \times \left( \frac{12}{24} \right)$$
Solving for $C$ since its constant:
$$C = 273 / \sin(\theta_1) / 0.416$$
$$C = 298 / \sin(\theta_2) / 0.5$$
Note that using similar triangles:
$$\tan^{-1} \left( \frac{\text{Radius}}{\text{Distance}} \right) = \theta$$
So:
$$\theta_2 = \tan^{-1} ( 15,000 / ( (15,000 / \tan(\theta_1)) -15,000 ) )$$
Where $\theta_1$ is the angle from the North edge and $\theta_2$ is the angle at the Center.
To achieve room temp at the middle of the plane we need:
$$298 = \sin(\theta_2) \times C \times 0.5$$
To ensure a proper temperature in the North the value of $C$ must be:
$$C = 273 / \sin(\theta_1) / 0.416$$
Combining with the proper temperature for the center:
$$298 = \sin(tan^{-1}( 15,000 / ( (15,000/tan(\theta))-15,000) ) ) \times 273 / \sin(\theta) / 0.416 \times 0.5$$
Solving for $\theta$$:
$$\theta = 0.785398°$$
Solving for Sun distance using similar triangles:
$$\text{Distance} = 15,000 / \tan(0.785398) = 1,094,200.5\text{ miles}$$
Using similar triangles again to get the angles for the North, Center and South:
$$\tan^{-1} \left( \frac{\text{Radius}}{\text{Distance}} \right) = \theta$$
$$\tan^{-1} \left( \frac{15,000}{1,094,200.5} \right) = \text{North} = 0.785397979°$$
$$\tan^{-1} \left( \frac{15,000}{1,094,200.5 - 15,000} \right) = \text{Center} = 0.79631297°$$
$$\tan^{-1} \left( \frac{15,000}{1,094,200.5 - 30,000} \right) = \text{South} = 0.807535595°$$
Solving for the appropriate $\text{Solar Constant}$ to get a 25 °C Center:
$$C = 298 / \sin(0.79631297) / 0.5 = 42,884$$
Testing out our temperatures for the $\text{North}$, $\text{Center}$, and $\text{South}$:
$$\sin(0.785397979) \times 42,884 \times 0.416 = 244 K$$
$$\sin(0.79631297) \times 42,884 \times 0.5 = 298 K$$
$$\sin(0.807535595) \times 42,884 \times 0.5 = 302 K$$
So the extremely low angle is less than 1° high in the sky.
Note that the South will have increased heat due to the rain shadow and the North is freaking cold and actually colder than that because the ice will reflect more.
