1)& 2) Yes , the maps are replayable (except for 4-4, which when you beat it completes the side story and you have to start again from the beginning: Raspberyl 1-1)
3) Yep, actually if you use the code to unlock it early they gradually level up through the story as well.
4) 4-1 or 4-2 I don't remember which off hand, its easily recognizable though, its the stage with about 7 Chearleaders standing on top of a huge wall in the middle of the map.
5) You need to complete Battle for the Test and Super Delinquent to unlock Culture Fair, completing Culture Fair unlocks Majiin Academy. Raspberyl mode is independent of the others.
6) No, a bunch of what were DLC characters in the PS3 version are unlocked with each "Chapter" of side stories you complete. I don't remember which one unlocks GIg in particular, I think it was Battle for the Test (which only has 1 chapter). You don't recruit them immediately , it just unlocks a Post-Game Map in which you have to beat a level 100 version of them to recruit them. The only characters you directly recruit during the side stories are SuperHero Aurum (unlocked by completing Raspbery mode in the post game) and Rutile and Stella who are unlocked after Majin Academy 1-1 and completing Majin Academy respectively