As noted in
How can a quasar be 29 billion light-years away from Earth if Big Bang happened only 13.8 billion years ago?,
the wiki about quasars still contains the following misleading sentence:
"The highest redshift quasar known (as of June 2011) is ULAS_J1120+0641, with a redshift of 7.085, which corresponds to a proper distance of approximately 29 billion light-years from Earth."
But even if the "strange 29 billion" is replaced by the "correct 12.9 billion", the fact remains that the actual measurement is "a redshift of 7.085". The "proper distance" is only a different way to express that measurement. It's not clear to me how "accurately" this describes the distance of the quasar, because the quasar surrounds a black hole and rotates quite fast. So there are at least two additional sources for the redshift, but how significant is their contribution?
Best Answer
You may gain some intuition on cosmological distances by exploring cosmocalc which gives a light travel time of 12.9 Gyrs for the redshift of 7. This site is nice because it shows you the parameters for the "concordant" cosmology that describe our Universe. These parameters describe how the universe is expanding and how it expanded in the past. One cool thing to try out is varying the redshift and seeing what ages for the universe you get at that redshift. This gives you a feeling for how a big change in redshift at high $z$ gives you little change in the age of the universe. For example a change from $z=7$ to $z=8$. But this represents a pretty big shift in the spectrum (challenging detection). A decade ago astronomers did not know whether the universe was flat or what $\Omega_M$ was (density in units of a flat universe). But now there are a number of different astronomical measurements that have all agreed on (or given small error bars around) these parameters. Hence the term "concordant."
To answer your question, a light travel time is a proxy for distance, it is an integrated quantity and so a meaningful metric for distance. However you can't really send a message back as the universe has been expanding since the quasar emitted the light that we are detecting. The second part of your question concerned accuracy. Probably the most meaningful error is the error in the estimated redshift. As you mention the emission lines can be broad and this would contribute to the error. The emission lines could be misidentified though since there is a Lyman alpha forest giving absorption lines from material in between us and the quasar this is unlikely for this object. The quasar could be moving w.r.t to the `Hubble flow' or mean expansion of the universe. For example the quasar could be a black hole that ejected from a galaxy. This type of error is likely to be less than the speed of light and so would at most contribute something like an error of 1 to the redshift (and that would be a really big error and so it's probably less than this). Likewise the emission lines could be broad but at most giving an error of about this size. At most. You can turn an error in redshift into an error in the frequency or wavelength shift of the spectrum. Even for low resolution spectra an error of 0.1 in $z$ would be just enormous. Errors in the concordant cosmology parameters would also affect the estimated light travel time. These days these parameters have errors of a few percent or so, -- using the cosmocalc you can figure out what affect variations in these ($\Omega_M$, $\Lambda$, and the hubble constant) would have on the light travel time.