Is it common to switch between natural, harmonic, and melodic minor scales when playing the same piece or is it best to stick with one?
From a "Common Practice" point of view, there is only the minor mode; there is also only a single minor scale but two of the notes are mutable. The sixth and seventh notes of the minor scale can be either the normal (agrees with the key signature) or raised. Which note is used depends mostly on local harmonic or melodic action or the whim of the composer. There are a few common practices but these are often violated for artistic reasons.
For melodies. (I'll use b and # for the two forms of notes 6 and 7 regardless of the actual signature used. Also normal and raised or flat and sharp. There's not a good terminology.) If the prevailing harmony is dominant, the raised forms of notes 6 and 7 are used. If the prevailing harmony is subdominant, the normal forms of notes 6 and 7 are used. If the prevailing harmony is tonic, the raised forms of notes 6 and 7 are preferred in ascending melodic lines and the normal forms in descending melodic lines. Sometimes, especially in instrumental music, the normal form of note 6 and raised from of note 7 is used. An arpeggio of a dominant ninth chord uses notes 4,5,6,and the #7 and 2 (or 9). The upper neighbor of 5 is the flat form of 6 as this makes a semitone and is much stronger sounding than the raised form.
For harmony. The major form of the dominant chord is preferred at cadences; the minor chord is common other places. The major IV chord does occur but not as often as the iv chord. The ii0 is common but ii minor and II do occur, the latter acting as a secondary dominant. The augmented III chord is rare except as a passing harmony The VII chord occurs often (and acts a bit like a subdominant) as does the #vii0 which acts similarly to the vii0 in a major key.
The practice of using various minor scales isn't common in the classical era; it's a more recent viewpoint. It doesn't much matter how one derives things if they sound good.
It depends to a degree on what genre of music it is, but it's not uncommon to use the notes from the different minor scales. In early music, the melodic was favoured, going up in melody one way, and using slightly different notes if the melody descended.
Some pieces use solely the harmonic minor notes, and some only the natural minor notes. In fact, for all three, the first five notes in the scale are the same anyway.
There are also modes which work in a minor fashion - the Aeolian is the natural minor, but there's also the Dorian and Phrygian, all containing that minor third from the root, making them ideal for a minor flavoured piece.
Jazzers seem to like the melodic minor scale notes, but use the ascending set only.
Well, it depends on whether you are playing melody or harmony, obviously. It's also pretty common for melodic instrument to play melodic minor on ascending scales and natural minor on descending scales.
Whether it's "best to stick with one" very much depends on the character of the piece and the intent of the interpreter.
The original of "Greensleeves" is a bit of a mixture of natural and melodic minor but it maps pretty well to the Dorian mode, and when you are playing it modally, sticking with the scale (and the respective harmonizations) makes for a more consistent character.
But if you are not after that sort of medieval or renaissance character, it's rather more than less common to vary according to local criteria.