We prove that a generic element in the Anosov-Katok class of the torus $\overline{\mathcal{O}}^\infty(\mathbb{T}2)$ acts parabolically and non-properly on the fine curve graph $\mathcal{C}^\dagger(\mathbb{T}2)$. Additionally, we show that a generic element of $\overline{\mathcal{O}}^\infty(\mathbb{T}2)$ admits sublinear rotation sets of all homothety types of non-empty point-symmetric compact convex subsets of the plane.