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
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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
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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
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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.