Character and game design are intertwined. It's the tone, the context, the character reactions all help define the character. DS and DmC have a similar mission structure (in that there is a guy who gives you a mission and you go do it), but in one case the mission giver is the one that comes off as the weaker person and in the other is the mission doer. Death clearly doesn't want to do any of this bullshit, and the people who give him the missions, even the ones weaker than him, are more or less unafraid of him and the tasks they give him are usually errands that they need done, but are clearly not dependant on Death to do them. By contrast, DmC has Dante, the one who goes out on missions, be painted as the more stronger one because Vergil clearly needs Dante and the world would be doomed if Dante didn't bother himself to help him out.
Also, if you seperate such a line between game design and character, then a lot of Vergil's weakness is justified. He didn't actually need to be saved while in that car mission, that was just the mission design. He totally could have beaten Mundus, it was just the mission design that had him get captured.
As for Lore, I think that DS2 also hurt that quite a bit, but I forget if that was because of Death or something else.
I'm not sure what that means. Because Death says badass lines in his private time, he is badass? DLC Vergil won't have trouble winning you over then. Death has an awesome visual design and has a good voice actor and the game plays fun, but you can't make excuses for him like this.