The quests with Sophia involve you coming to terms with her backwards speak. As such, I don't think you ever reveal your wooly form to her outside of the Unity Festival, but I could be wrong.
That being said, the request section of the FAQ/Walkthrough does a good job of pointing out when and where you can complete what quest without spoiling everything. If you can't get the next request until after the Unity Festival, the answer is definitely "no". Other than that, wait a few days, and rotate between the mailbox and the billboard if the next quest in line isn't showing up.
Nope. You cannot reveal your Wooly form to these characters until the Unity Festival. They only have three quests, and for the most part, they're not anywhere near as serious as the nine bachelorette quests. Sakuya's quests actually get put on hold until after the Unity Festival, because the end quests rely on Shino knowing Micah's secret.