For question 2, the original manifold M will be a double cover of the quotient space since it's an involution, and if you have an operator that commutes with the involution on the covering space, it should descend to the quotient space. If there are "nice" fixed points one can get an orbifold or a stratified space, where you can still make sense of a Dirac operator and all the structure, possibly after resolving the space. For instance https://arxiv.org/pdf/1709.01636.pdf https://arxiv.org/pdf/1712.08513.pdf

For question 2, the original manifold M will be a double cover of the quotient space since it's an involution, and if you have an operator that commutes with the involution on the covering space, it should descend to the quotient space. If there are "nice" fixed points one can get an orbifold or a stratified space, where you can still make sense of a Dirac operator and all the structure, possibly after resolving the space. For instance Hartmann, Lesch, and Vertman - On the domain of Dirac and Laplace type operators on stratified spaces and Albin and Gell-Redman - The index formula for families of Dirac type operators on pseudomanifolds.