Confession time: the first time when I played figured bass (I was very young) I just put chord symbols in my score. It might have even been the same concert where we played French Baroque with notes inégales and the bassist just put “Swing it!” on his music.
That would depend whether the “6” is intended to signify an added 6th or the first inversion of the D chord. I think most would interpret it as an added sixth.
Perhaps I misunderstood the original frame of reference. I was think of chord symbols rather than figured bass: D6 or II6. I would interpret a 6 under a D bass note as VII6 or the way @PjotrB did.
It is, but usually the bass note is not being played in the 6-chord (right hand), it’s 6/3 rather than 8/6/3.
In contrast the bass note is being played in a 6/4 chord
In the realm of figured bass, probably not. The 6 is, as others have clarified, shorthand for 6/3. If the composer wanted a Dm6 chord, the figures would have to be 6/5 (shorthand for 6/5/3).
It is really a very practical thing rather than a theoretical; D is in the bass line, and instead of just a standard triad 5/3 I play 6/3 instead; as a figured bass player I do not even have to think about inversions and roots.