This came up in a local study group for L2 candidates, and I had a lot of difficulty finding CR references to back these up.

I know that you can target a split card in the graveyard with GD-D if one side is CMC < 4 and pick either side, because the CMC comparison returns two values, one of which is positive. (708.5).

How is the legality of the spell on the stack checked once we decide to pick the greater side? Is 601.3 relevant here?
If we look at the card as last known in the graveyard, how does this compare with the Expertise cycle when we don't have the card in a public zone before it is put on the stack?

601.3 is not particularly relevant here. You needed permission to cast the card in order to begin casting it, and you got that permission from Goblin Dark-Dwellers. That's all that matters. Effects such as Goblin Dark-Dwellers and the Expertise cycle work by pointing to a particular card and saying “You have permission to cast that card”–they don't care at all about the characteristics of whatever spell may result from you casting the card.

To give an analogy, imagine I have a variety of toys of various colors, and tell a child “Pick a blue toy to play with.” The child picks one. If while playing with that toy the child happens to dip it in red paint, the toy not being blue any more does not mean they no longer have permission to play with it.