I'm attempting to create a world that has very brutal, hard swinging weather and environments, leaving much of the planet desert-like. The world I've designed is a moon of a gas giant.

The Parent Planet

This planet is a gas giant orbiting a sun smaller and cooler than our own. It's closer than the goldilocks zone of this star, so it is quite warm. It has a wide band of rings, and a single moon orbiting closer to the planet than the rings.

The Moon World

The moon is earth sized, not tidally locked, and experiences days similar in length to earth days, and its orbit is about 40 of these days. Most of the year, due to the rings, this planet has a band at the equator of essentially arctic tundra, never seeing sunlight. However, due to the axial tilt of the parent planet, and the moon's slightly erratic orbit, each hemisphere spends a quarter of the year (Each year being about four orbits, or 160 days) more and more in the shadow of the rings, making for one very, very cold orbit (-40F, -40C at night). On the other side of the year, with one hemisphere out of the shadow of the rings entirely, the surface becomes blazing hot, reaching regular temperatures of 120F (49C) During the day. During the other two orbits, with some shadow from the outer rings, temperatures are more stable and around 80-60F (27-16C) high and 60-40F (16-4C) low.

Question

Is this proposed moon possible? Would it act how I expect it to? If not, can I fix it? Bonus points for pointing out any cool and unexpected visual effects.

(In answer to the original question: Is it possible to have a planetary moon closer than rings?)

It doesn't really make sense.

Rings and moons aren't unrelated features that just happen anywhere around a planet. Rings are what happen when a moon is too close to its parent planet, i.e., when it is inside its Roche Limit

It's theoretically possible, if you had a really, really dense moon (e.g., pure osmium) orbiting just inside the rings that resulted from a really, really light moon (water ice), but it would be a very contrived situation.

So... the moon's existence might be possible given the planet's own position and composition. But for the smaller details like the tidal lock and the temperature, I'm not so sure about.

Planets form from leftover debris from the formation of a star; rocks and solids usually orbit closer to the star as further out the gravitational field would be too weak to lock them in orbit. However, gas giants can only form within the larger clouds of gas and ice further out as only there are the materials abundant enough to make them 'giant'; like the gas giants within our own solar system.

There would have had to be some kind of push from another celestial body to nudge it that close to the star.
And yeah, there's a precedent for this happening:
http://hubblesite.org/hubble_discoveries/discovering_planets_beyond/how-do-planets-form

So now you need a large celestial body to give a gravitational nudge to the planet. And there you might be able to explain some of the rings; some pieces from a nearby asteroid belt were thrown away by its gravitational field and nudged it ever closer to the star while others were trapped in the field and began orbiting it as rings.

The moon could be what's called a "Shepherd Moon"-- one that orbits in the very inner or outer circle of the rings. After some more research I don't think it's possible for a moon to form any closer than the shepherd moons in the rings (which keep them from clumping and collapsing onto the planet), there's a lot that can happen in space and the possibility it could occur might be very unlikely but still possible with some extreme circumstances and a bit of glossing over.
https://www.iflscience.com/space/how-saturns-shepherd-moons-herd-its-rings/
https://physics.stackexchange.com/questions/26643/why-arent-saturns-rings-clumping-into-moons
https://en.wikipedia.org/wiki/Ring_system

It's not the best explanation, but this could suffice as a placeholder until something better comes along.

There is no 'shadow of the rings'

Saturn's majestic A ring is 10-30 meters thick; overall the rings of that planet range from 10 meters to 1 km thick.

The asteroid belt is a 'ring' around the Sun. That, too, is not dense enough to block the sun. In this answer, I show that if the asteroid belt was ground down into particles 100g in mass, each particle would have 1530 km$^2$ of space to itself; each particle would be about 14 km from the next particle.

Rings just aren't thick enough to create an appreciable shadow.