Questions about worldbuilding and storms on a ringed planet

[Andrej Butić]

Hey friends, I've been working on a book for a while, and in it I have a fictional earth-like planet with a system of rings (like Saturn, but narrower), and I've been thinking about the effects this might have on the atmosphere.

More specifically, every winter, the sun will at some point, be blocked out by the rings, causing a sudden cooling in the region under the shadow. If the rings are narrow or the latitude low, then the sun might be able to slip behind the rings and back within one day. On higher latitudes or with wider rings, a portion of the horizon will be covered by the ring, potentially causing a "polar night" at much lower latitudes, with only a fraction of the sunlight poking through. This effect can be seen in the following image of Saturn: 

I know the weather variables with such a system are probably insane, so I'll try and condense this into a few questions.

Say it's a cloudy, but warm/humid day. The sun passes behind the ring, and suddenly, the upper clouds/upper atmosphere are in shadow, and experience rapid cooling. Could the temperature difference above and below the cloud be enough to cause a storm?

Say it's a clear, cloudless day, and the sun passes behind the ring, what will you experience? A sudden cold and increased wind speeds?

The ring forms a shadow band that shifts up and down the planet throughout the year. Will the weather be wild in and around the edges of this band?
 

Since the amount of light that can get blocked by a ring system is also a variable, I think it's safe to assume that the effects will be much milder if the rings block out 30% of the light as opposed to 90%