This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations {are} established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that $H^γ(\mathcal{O})$ is compactly embedded in $L^2(\mathcal{O})$ with $γ\in (0,1)$. Moreover, the uniqueness of this invariant measure is presented which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of SPDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations