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Suppose a gas giant is incredibly large with a super dense, solid core that hold the gas around it like an atmosphere.
Would small moons be able to exist within the gas, assuming the gas has little effect on the moon's and their orbit around the dense core. Also, what type of life, if any would be able to exist?
Not for long they wouldn't.
Within the atmosphere they would be subject to air resistance. Air resistance means drag, and drag means that, in the absence of some force to maintain its momentum (do your moons have giant rockets with unlimited fuel?), the moon will lose speed. Less speed means it can't maintain its orbit, so it falls lower, where the atmosphere is more dense, where there's more drag, so it slows down ever further...
You can see where this is going.
In spaceflight this is called "aerobraking"; in the absence of flight control surfaces, parachutes, air bags, or similar means of controlling one's descent or cushioning one's impact, the inevitable conclusion is "catastrophic lithobraking".
As Kromey explained, an actual orbit would quickly come to an end. However, if the lower atmosphere is more dense than the moons, and the upper atmosphere is less dense, then the moons will float inside of it. If the wind in the atmosphere is regular enough, you could get something very similar to an orbit. The moons would have to have a very low density, or the lowest part of the atmosphere would have to have a very high density, though.
In most cases, the thickness of the atmosphere would mean that life on such moons would be veiled in darkness. Not being able to get energy from the sun, the first life might rely on friction from the atmosphere to generate heat. For life on such a moon to be water-based, the atmosphere would have to contain water, meaning the moon would have to be about as dense as that water, or the moon would have to contain water or ice.
