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I was wondering a few things

First, their successors retaining very close (apparently legion-like) links to the original DAs...Now, does this include just the successor chapters or every DA derived chapter?

And second, does "The Fallen" mean just the dark angels who sided with Luther or does it apply to every Dark Angel who has turned to chaos?

All DA-derived Chapters are, technically, Successor Chapters. This is true of any Chapter descended from any other First Founding Chapter. Whether they know it or not, they are a Successor to one of the original Legions.

The Fallen are, specifically, those who sided with Luther/fell to Chaos during the Heresy (or failed to fall to Chaos and follow the Lion, depending on which story you believe). More-recent traitors might be lumped into that bunch, but we're not given a lot of details on that situation/classification. Given that the Fallen are set apart from regular Chaos foes, though, I don't believe that it includes any more recent traitors.

You aren't cleared for that information. Report to the chaplains for re-education.

(but Psi's right).