Call of Duty: Modern Warfare II does support crossplay on PlayStation 4, PS5, Xbox One, Xbox Series X, and PC (via Steam and Battle).

Yes, all of the Call of Duty: Modern Warfare games on PS3 have a singleplayer campaign. Some of the campaign modes have a co-op mode, but I'm not sure which of the games has this. Call of Duty 4: Modern Warfare has singleplayer and multiplayer.