It's a bit of a pipedream, but something like a DMC.net in the vein of Re.net for overall stat tracking, weekly events, and future unlockables would be great for the series and community.
Multiple playable characters, even if unrelated to the canon story, are a must, whether included in the campaign or designated as DLC for missions. This plays hand in hand with the proposed coop play.
Also, I think elaborating on Vergil 3 is important, and why that particular boss fight is so strong. It's largely due to the pacing of the game and the buildup from the prior fights. There's so much going on with that particular example. But on a base level, you could just request more humanoid fights overall like Vergil and Credo.
Also, including "legacy" boss fights as DLC would be nice additions to mission mode. Bring back favorites like Phantom and Nelo Angelo in a new setting. Let us fight Vergil+Nelo Angelo at the same time.
Along the lines of no Sparda wannabe, I'm going to suggest that they need a new and charming antagonist. No lame businessman. No old pope. Give us new blood (or resurrected old blood) for Dante and co to fight.
There's some debate over this, but I'm partial to a hub or map world mission Select screen that would be your customizable Devil May Cry shop. From there, you can select different stages in a different order, so there are plenty of side missions to select from... and only a few missions will progress the story to unlock more story related missions. Upon beating a stage, you can replay it for a better score or with a different character.