Embedded doubly periodic surfaces of genus 0 have been classified by Hippolyte Lazard-Holy and William H. Meeks in 2001. For genus 0, the case of parallel top and bottom ends has been classified in 2005 by Joaquín Pérez, M. Magdalena Rodríguez and Martin Traizet. For the non-parallel case, it is conjectured that there is only a 1-parameter family, with the Karcher-Scherk torus as the most symmetric example.
It is possible to add a handle to the doubly periodic Scherk surface if tilt the (say) bottom ends. These surfaces won’t be embedded, and attempting to make the bottom ends parallel will move the handle into the ends.
Despite not being embedded, they are nevertheless interesting, as they arise as dihedralization limits of a singly periodic Scherk surface with exotically placed handles.
Resources
Mathematica Notebook (by Ramazan Yol)
Minimal Surfaces 