There are three basic functions a chord can have in a song: Tonic Function, Subdominant function, or Dominant Function. They get their functions from their numerical position in a key.
- Tonic Function (stability) I, III, VI
- Subdominant Function (contrast) II, IV
- Dominant Function (tension) V, VII
This works in Major or Minor, A III chord and a bIII chord are both tonic function chords. In Rock and Blues I, IV, V are the three most important, in Jazz I, II, and V are.
The bVII chord (G or G7 in A minor) does create tension that can be resoved by whole-step to the tonic, but its not as strong as V to I.
The bVI chord (the F in A minor) is normally a Tonic function chord, however you are correct it can take on the Subdominant function, usually when moving to the V (E). The easiest way to wrap your head around it is to think of it as a tritone substitution for the Subdominant II chord.
Given a basic progression you can create some variation in the chord progression for swapping out a chord that has the same function. The reason this works is because they share many of the same notes, and this process is often called Common Tone Substitution for that reason.
It is common in minor harmony to change the qualities of the chords as well. The normally minor V- chord is almost always changed to a dominant V7 or major to create stronger resolution to the I-. It is also common to change all the chords except the root chord to Major chords. House of the Rising Sun is a good example. You can also change all the chords to Major chords, from the perspective of popular music, it remains minor as long as the roots of the chords in the song, together add up to a minor scale. This is how rock and metal songs played only with power chords manage to be Major or Minor.
Here it is laid out in a table on my website:
Chord Functions and the Uppercase Roman Numeral System