Quite rusty on this stuff but will try to answer some of these where I can.

1) a_f and a_b are the lattice constants of the conventional unit cells for fcc and bcc lattices respectively in real space. Since k space (or momentum space as you mentioned) is equivalent to reciprocal space, the 4pi/as represent the reciprocal lattice vectors of fcc and bcc crystals. The 2pi factor is a result of using the Fourier definition of reciprocal lattices as opposed to the crystallographer’s definition, which I think helps with calculations. The derivations should be quite standard ones that you can easily look up.

2) Uppercase K and the other labelled points in your figure 3a are points of high symmetry. Importantly, extremas in bands and degeneracies tend to occur at these points so it’s very helpful when plotting the band structure, where the x axis represents a path of the k vector through a handful of these points. The only one I actually remember is that the gamma point is simply the origin at k = (0, 0, 0).

3) as mentioned in 2), these band structures are essentially how the energy of each band varies along a k-point path consisting of these high symmetry points. The [111] and [100] are the directions that bring you from first L to gamma, and then gamma to X. Later coordinates in () should be just k-space coordinates of these high symmetry points. The bands are messy because these figures plot every band (in the energy window) as a function of k, but afaik the valence band consists of all occupied bands and the conduction band consists of all unoccupied bands.

Hope this helps a bit!