NavList:

The horizon mirror has no adjustments, correct, but this is less critical to performance than one might suspect at first.  Essentially one can choose between side error and perpendicularity error using the single index-mirror screw (leaving aside the nearly-independent index-error adjustment, which rounds out the available suite of two adjustments).  On mine I adjust for perpendicularity (of the index mirror to the plane of the arc), and let side error fall where it may.  The error for sight angles more than a few degrees is negligibly small.

For sextant adjustment theory, I'm not sure there's anything better than the old books by Chauvenet, "A Manual of Spherical and Practical Astronomy" (in two volumes; the second volume has the particulars on sextants), and Simms, "The Sextant and its Applications".  Google has full PDFs for download.

An artificial horizon is very useful for practice on land.  The eyepiece shades on the Mark 3 are not strong enough to attenuate the Sun's light to safe levels, even with a liquid flat (let alone a mirror), so you'd need to supplement with a stronger shade such as a piece of eclipse-glasses film.  Or pick a landmark near the horizon that's reasonably near the Sun's azimuth and pretend it's the true horizon; that's fine for practice too.

You can test the sextant, and maybe make a small table of corrections for arc error, using terrestrial angles, if those are measured with a theodolite first.  Alternatively, star-to-star angles can be used.

Despite its limitations, the Mark 3 is fine for honest navigation---positioning to within a few nautical miles is within its performance envelope with careful use.  The only thing it really struggles with is lunar distances, which need higher accuracy.

Cheers,

Peter