Working on a farm is rough and dirty, and it's understandable she wouldn't want her father's final gift to be tarnished. So she instead wears hats of the same style to still remember him, but they don't have the same sentimental value so it matters less when they get messed up. She wore her father's hat for Hearth's Warming as a way to have him be there for the holidays when he couldn't physically be there.
He was going to get it for Fluttershy as a gift anyway, but realized he'd probably land in hot water with her friends if he brought it. So he used Rainbow, believing they would be more forgiving of her.
Fluttershy's conversation with Rainbow sounds a little teasing/trolling. She had planned with Discord beforehand, the end result being that both of them got to mess with Rainbow, and Fluttershy got a new animal friend she wanted- the winterchilla. Discord's gift of making her the hero was his own idea, one he hadn't told her. Holly the Hearthswarmer wasn't part of the plan- Fluttershy felt the first conversation was enough messing with her friend for her.
This in particular is the first occasion, and he is visiting Mount Aris for the Three Days of Freedom Festival.
- It would make the most sense as he wasn't with the Yaks nor the Changelings in the end, and the Dragons probably wouldn't be a fan of having non-dragons hanging around their lands.