How are the compositional components of exoplanet atmospheres differentiated? How are exoplanetary atmosphere compositional spectra distinguished from those of the parent star(s), from the composition of the planetary surface or any other factor?  Is it actually possible to determine the atmospheric composition precisely using this method?

Are there any specific examples of such an analysis having been performed on an exoplanet?
In theory one should just be able to determine the difference between the spectra during the star's eclipse of the exoplanet (starlight alone) and the spectra of the star and exoplanet together, but in reality equipment is not precise enough for this. To remedy this problem, the analysis is integrated over many eclipses. Some other calibrations are made as well which you can read more about in the paper A Spitzer Spectrum of the Exoplanet HD 189733b. This is called occultation spectroscopy.
There's another method used as well called transmission spectroscopy, which detects the change in light as the exoplanet passes in front of its star. You can read more about that in this paper: Infrared transmission spectroscopy of the exoplanets HD 209458b and XO-1b using the wide field Camera-3 on the Hubble Space Telescope.