You should first of all be aware of the fact that your system is not periodic in C, and thus you can't really obtain any band structure. Second, there is no immediately suitable built-in orthogonal Slater-Koster model in ATK for this problem, but you can try some of the DFTB model you list. To know which one is more suitable you must test them on something reasonable where you can compare to experiments or other theoretical results, like the band structure of a basic nanoribbon for instance. The default boundary conditions will generally be fine, unless you plan to add gates. In general, please read our tutorials, all points you ask are basically explained and usually in a tutorial for graphene...
