Am I safe in assuming that's 100% out of 104%? I haven't talked to anyone that has the Mask of the Ancients dossier yet, so I'm guessing IceWater doesn't have everything completed. Well, currently you can't be at full completion since there's no way of getting Astral Tiki and Lion until they're available for purchase.
I'm currently sitting at 101%. All I have left from the characters is four Shin Hisako dossiers, Mask of the Ancients, Crimson Owl and the Astral Tiki / Lion stuff. So I think about ten to twelve dossiers in total. Would LOVE it if someone was finally able to get Mask of the Ancients so I could find out how to get it.
There is no fast way. You just have to play. I'd suggest trying to gear who you actually play against toward characters whose dossiers you need.
No, at least not at the beginning (or middle). Although the missions certainly help, if you're just starting out, you want to play everyone you can as much as you can. Don't start narrowing down until you need to.
Doesn't seem overly relevant as far as I've seen. Shadows, Mimics, and real characters can all drop dossiers. I have yet to see any of them drop dossiers at a greater rate than any of the others. Though certain characters do tend to drop dossiers more frequently than others. Some characters, Omen, TJ Combo, Jago, Aria and Cinder seemed to be harder for me to come by than some of the others, but that was just my experience.
All of them. Switch it up more in the beginning, but then as you start to need specific dossiers, start using character that have missions that tend to drop dossiers more often. If you need Omen dossiers, use Jago more (for example). I believe Mira and Maya both tend to help get one another's dossiers, if I'm not mistaken.