Alright, time to drop some fanfic on you guys.
Personally, whether it was intentional or not, Argosax the Chaos in his Despair Embodied form has wings that look ridiculously similar to the transparent wings around Sanctus in his demon form in DMC4. In DMC4, Sanctus achieves his demon form by bonding himself with the soul of demon. I think that overall, it'd add alot more to the story if Argosax is the demon that Sanctus chooses to bond with. Seeing Sanctus as a pawn to break into the Human World, Argosax is allowed by Sanctus to posses him, not knowing his true intentions. Using Sanctus's plan, in order to gain the power of Sparda, Argosax would pose an incredible threat to the Human World. Thus, Dante's seemingly random appearance on the scene expands far past the weak "Lady Job" storyline we were given. Finally, Argosax would present the greatest threat to the Human World yet, and we'd all have a badass baddie to root against. Ultimately, Argosax is unable to cross over, after an unholy trouncing by Nero. Sanctus is only human and therefore unable to fully wield Sparda's power, a failure shared by Arkham in DMC3, and thus is easily defeated. Argosax is unable to cross over with the power of Sparda, and the day is saved.
Now
DMC2 starts with the pretext of Dante seeking out the defeated Argosax on Vie de Marlie by assembling the Arcanas, powerful relics with enough power to allow Argosax to cross over. After witnessing the devastation brought by Argosax to Fortuna, seeing Argosax finally brought to justice becomes Dante's main motivation. Once there, Dante encounters Argosax's newest pawn, Arius. Arius of course would be an obvious choice, considering the similarities between him and Argosax's last pawn, Sanctus: High position of power, influence, greed, etc. So, eventually the story jumps through about the same hoops and Dante has a showdown with Arius over Lucia. When Dante throws in his coin, along with the rest of the Arcanas when he presents them to Arius, it gives Dante the opening he needs to jump in and face the elusive Argosax Face-To-Face! Additionally, should he fail, the Hell Gate is unstable enough to remain open for long, ensuring that either way, Argosax will not cross over. This of course does two things. The first of which, is it provides Dante the forward momentum and clarity in script that DMC2 sorely lacks. I mean cmon. ... it'd be just like Dante to literally throw Argosax into a room, close the door, and duke it out man to man. And two, it gives all of the characters, especially Arius and Argosax, a greater and more fullfilling purpose within the storyline.
As far as Dante's attitude, it'd irritate me to have to work in DMC2's script writing failures into the final story. Keep Dante as the same ol' Dante we know and love. Sex up the weapons. Sex up the fighting style. Jesus christ, sex up the music. Sex up the environments (As Lexy suggested). Take everything ****ty about DMC2 and leave it DRIPPING with AWESOME. Anyway, with that in mind, that's how I would retell DMC2. So essentially, this would not only make DMC2 way better, but it would take the seemingly pointless "monster-of-the-week" story of DMC4 and add a sense of granduer and purpose as well, uniting the two weakest Devil May Cry games (4 and 2) and giving them more to bring to the overall series's canon. Keep in mind, this is all accomplished, not by rewriting the story, but by retelling it in the context of the existing story. In the end, the current games kind of already tell this story, although poorly and unstated. Anyway, that's how I'd do it.
Side-note, I might be one of the only ones who didn't mind Nero's backstory not being expanded upon. I was really ticked off at first, additionally with the fact that Vergil didn't resurface in DMC4, but I think all of these questions would be better answered in another game. With the addition that Nero is just a potential treasure trove full of possible script ideas, the most important thing that needed to happen in DMC4, is the introduction of Nero into the series canon, and his encounter with Dante, which was definitely accomplished. As far as Nero's non-involvement in DMC2, I think Nero would leap at the opportunity to join Dante and the Devil May Cry to fight demons. But having fought demons his entire life, i feel that Dante would reject Nero's offer, in hopes that Nero and Kyrie can have a normal life (La Vita Nuova) without strife or conflict. This of course sets up a unique character dynamic between Nero and Dante. Thus, when Dante goes missing and is trapped in the Underworld, with the addition of Dante's eventual return, I feel like there's endless story possiblities to further flesh out the interesting and strained relationship Dante and Nero would realistically have. But now I'm getting way off topic ...