I'm not sure exactly what your header means, but maybe I can suggest partial answers to your questions. First of all, there are by now many ways to approach the original Borel-Weil theorem, depending on what machinery you are inclined to use. While it can be formulated in several related settings, this theorem basically provides a model for the finite dimensional highest weight representations of a semisimple algebraic or Lie group in characteristic 0: take global sections of a suitable line bundle on the full flag variety $G/B$.
Bott and Kostant both explored some of the same territory but in different ways.
Bott's 1957 Annals paper extended the Borel-Weil theorem to other line bundles on $G/B$ and was both analytic and geometric in spirit, but as a consequence he observed that it was possible to derive the dimensions of certain Lie algebra cohomology groups. Here the flag variety corresponds indirectly to the nilradical of an opposite Borel subalgebra in the corresponding semisimple Lie algebra. In Kostant's 1961 Annals paper, he worked out more directly some of the related Lie algebra cohomology in the context of Borel subalgebras and their nilradicals (and more generally in terms of parabolic subalgebras).
The BGG resolution of a simple finite dimensional representation of the Lie algebra later permitted a quick and transparent proof of the Bott-Kostant dimension theorem. The resolution also has a parabolic version, as in Lepowsky's 1977 paper
here and in Rocha-Caridi's Rutgers thesis supervised by Wallach. (Later work simplifies much of this, as recorded in my 2008 textbook on the BGG category: see 6.6 and 9.16.)
As indicated in the question, the original geometric setting of Borel-Weil and Bott was greatly simplified by Demazure in his 1976 paper here. This work relies on methods of algebraic geometry and only concerns the computations of line bundle cohomology on the flag variety, so it's not clear to me how to connect this directly with the Lie algebra cohomology. As far as I know, the only relevant literature (due to people like S. Kumar and O. Mathieu in the late 1980s) deals much more generally with Kac-Moody groups and Lie algebras. Maybe one can extract something from their work in the finite dimensional Lie algebra case which applies to the Bott-Kostant results.
[One side remark is that Henning Haahr Andersen, in his 1977 MIT thesis and in many papers thereafter, explored creatively in Demazure's style the more complicated cohomology of line bundles on flag varieties in prime characteristic. Parabolic subgroups often play a role, as for example here. But here one is much farther away from the setting in which Lie algebras play a prominent role.]
