Extreme Planetary Features vs realism

[Chick]

If a gas planet (like Saturn) migrated close (but not too close) to a star (maybe the size of Alpha Centuari), would it be viable for the moons of this gas planet to absorb gases from the planet similar to the companion effect that impacts stars? If the Gas Planet had plenty of Oxygen and Nitrogen, could it be possible for life to be sustained on these moons?

Not impossible if there were any reason for the gasses to migrate from the planet to the moon, but I can't think of why/how that would happen.

Could the atmosphere of a gas planet comprised of oxygen (primarily) extend beyond the moons, which if solid and contained enough water, produce or sustain life?

Impossible, due to Roche Limit, as Azelor said. http://en.wikipedia.org/wiki/Roche_limit

All of that said, there is no inherent reason that the moon of a gas giant couldn't contain an atmosphere, and if within the habitable zone of a star, liquid water and potential life.

Consider Titan, satellite of Saturn, which contains an atmosphere denser than Earth's, and Europa, satellite of Jupiter, which is suspected to contain liquid water beneath the ice surface (melted by gravitational friction due to Jupiter's proximity). If either of these were in the habitable zone of Sol, they might well be habitable.