When the light is analyzed from distant stars, do the Red-Shifts include atomic absorption lines being shifted too? Is the amount of Red-Shift measured, show differences in atomic absorption lines?

Yes - absorption lines are shifted if the star is moving relative to us. This is a good techniques for determining the relative motion of a start, But when you're looking at light from a galaxy that is far enough away to show appreciable red shift due to Hubble's constant you can't really pick out one star's spectra, because the galaxy is too far away to resolve a single star from the masses of other stars in the galaxy. So you tend to see a sort of averaging of spectra from many stars in that galaxy, and hence absorption lines can get smushed out a bit.