I've been working through this A minor scale and I just can't figure why the fret changes when I descend the scale — how is G sharp; now on a different fret when I'm doing it as A flat? Is the tab correct here? It doesn't seem right to me — surely going from A to A flat is one fret change and not two.
In the Common Practice Period (technical name for composers doing what I'm describing), the keys were (and still are) divided into two modes, major and minor. Each of the 12 (not counting enharmonic things like C#=Db) major keys consist of the usual 7-note major scale pattern: SSHSSSH (from C, DEFGABC). The minor modes have a flat third (Eb in the case of C) and mutable sixths and sevenths; that is, (in C minor), the sixth step can be A or Ab and the seventh step can be B or Bb.
The usual explanation is that composers like to approach the tonic (C for example) by a half-step. So in C minor, melodic motion often goes B-C rather than Bb-C. This also allows for a major dominant chord (as in major keys). Also, augmented seconds tend to be avoided melodically, so B is often approached from A rather from Ab. There are many exceptions, especially in instrumental works.
The raised scale steps tend to be used in ascending passages (CDEbFGABC) and the lowered steps in descending passages(CBbAbGFEbDC). The lowered third stays lowered. However, it's not unusual to have passage like ...GAbBC which sounds a bit exotic and also outlines (part of) a dominant ninth chord.
There are a few common practices such as, when using tonic-oriented harmony, one uses the raised steps in ascending passages and normal scale steps in descending passage. When using subdominant-oriented harmony, the normal scale steps are used in ascending and in descending passages. And when using dominant-oriented harmony, the raised sixth and seventh steps are preferred. These are only common practices and not absolute rules.
The upper neighbor of the fifth step is almost always the lowered sixth, rarely (I've never seen it.) the raised sixth.
The two accidental signs after the high A are natural signs. You play G-natural and F-natural when descending in the A Melodic-Minor Scale. The A will never be altered in any kind of A-scale.
More generally, the 6th and 7th scale degrees in a Melodic-Minor Scale are considered movable and may change through the course of a piece to suit the melody.
The fret markings are correct.
Just adding to the previous two answers, there are three types of minor scales:
In the example you've given, they're using the Melodic Minor form. In most music, the Natural Minor & the Harmonic Minor modes are commonly used but the Melodic is perhaps the least used out of the three modes. If I were you I would try to practice the Natural Minor & the Harmonic Minor if you can find them.
