How deep is the "A" Fraunhofer line in the solar spectrum? Is it from the Sun's or Earth's atmosphere?
Where can I see - or even download - a fairly high resolution (~0.1 nm) solar spectrum at zero air mass - in other words from space without absorption and other features from the Earth's atmosphere?


You can download a normalized solar spectrum with 0.1 nm resolution from here. However this is not at zero airmass.

A solar spectrum with zero airmass can be found on this website with the direct link to their data in an excel sheet here. This spectrum appears to be at a 1 nm resolution.


  Of particular interest is the near infrared around 750-780nm. Is the "A" Fraunhofer line due to absorption by oxygen in the Sun's atmosphere, or in Earth's?


If you look up the Fraunhofer A line (really it's a "band"), you'll see it is caused by molecular oxygen, namely $\mathrm{O_2}$. This should tell you right away that it is caused by our atmosphere as molecules do not easily exist in the atmosphere of the Sun. In our Sun's atmosphere, most species are ionized forms of single atoms such as $\mathrm{O\:IV}$.

In Astronomy, spectral lines which are caused by the Earth's atmosphere and not by the source object being observed, are referred to as Telluric lines (from the latin word tellus meaning Earth). I found this source which describes, in detail, the various telluric lines that you may encounter.