Solved
About EDI
In Mass Effect 1 Rogue VI you do not know it is EDI. You do not find out until Mass Effect 3 Cerberus Headquarters/Cronos Station. What would happen to EDI if you do not do the Rogue VI Mission besides not getting Specialization Class? Just asking.
[Edit: Holger1405 removed oversized text.]
If you mean in-game, the completion of Rogue VI affects some dialogue (see this part of the wiki for the scene in ME3 you mention specifically).
If you mean lore-wise, with EDI's origin staying the same regardless I think it's safe to assume that (like the outcome of the Arrival DLC) Hackett eventually sends an Alliance squad to complete the mission if Shepard doesn't do it themselves.
