I'm not an expert in this area, but I'm told that the key phrase in the superalgebra world is "Weyl groupoid" rather than Weyl group. I did not look at the construction long enough to understand it. Serganova has a paper describing foundations in a super analogue of the Kac-Moody setting, and you can find a description of the Weyl groupoid there.

I'm not an expert in this area, but I'm told that the key phrase in the superalgebra world is "Weyl groupoid" rather than Weyl group. I did not look at the construction long enough to understand it. Serganova has a paper describing foundations in a super analogue of the Kac-Moody setting, and you can find a description of the Weyl groupoid there.

Edit: The situation seems to be the following: For Kac-Moody algebras, there is a unique conjugacy class of Cartan subalgebra (under automorphisms), and the Weyl group acts transitively on systems of simple roots. These properties fail to hold in the superalgebra setting. One instead can form a groupoid whose objects are finite size square matrices $A$ with integer entries (or the Lie superalgebras $g(A)$ obtained by a generators-and-relations construction), and whose morphisms from $A$ to $A'$ are superalgebra isomorphisms $g(A) \to g(A')$ that take a Cartan of $g(A)$ to a Cartan of $g(A')$. The Weyl groupoid of $g(A)$ is then the connected component of $A$ in the larger groupoid.

Regarding geometry, I think Penkov has done some work with flag supermanifolds and Borel-Weil-Bott. I don't think there is much debate about the foundations of supermanifold theory, but I guess the geometric representation theory doesn't extend from the even case by rote translation of proofs.