With my first char, I made some quests in "One Person" mode, but I never started (nor finished) the quest "The Tinkerbell's Dog 2" (Side Story). After some time I noticed that it's marked as "finished" in my quest list (i.e., I can't start it, probably until I finish all other quests).
I used a second character to verify this. With this character, I never played any solo quests. I started "The Tinkerbell's Dog 2", finished it, and after that I was still able to play it again (and again).
Edit: With this second character, I noticed that she now has a "Main Story" quest finished (although she never did it): "Forest of Sorrow"
I used a second character to verify this. With this character, I never played any solo quests. I started "The Tinkerbell's Dog 2", finished it, and after that I was still able to play it again (and again).
Edit: With this second character, I noticed that she now has a "Main Story" quest finished (although she never did it): "Forest of Sorrow"