For the first question, there is another (paerhaps more conceptual) proof using the so called moment map of symplectic geometry. This proof, probably due to Souriau, is to consider the so-called moment map of this action. This map transform your manifold into a co-adjoint orbit of your semi-simple Lie group. One proves that this map is an isomorphism, not only for the symplectic, but also the complex structure. Souriau (Structure des systèmes dynamiques, Dunod, Paris, 1970, p 117. unfortunately written in a very high degree of generality).

For the second question, the proof I just sketched gives that your (non compact) manifold is a covering of a co-adjoint orbit. It appears that the fundamental group of the co-adjoint orbit, i.e. the orbit of a point under the co-adjoint action) is the set of connected components of the isotropy group is this point. Sometimes these objects are not simply connected. The proof that in the compact case, co-adjoint orbits are simply connected is not so easy.

For the last assertion, it is easy to make complicated non Kähler (hence non flag) compact complex examples. Consider the action of Sl(2,C) on C^2-(0,0), a plane without its origin. This action commute with homotheties, and therefore pass to the quotient in a transitive action of Sl(2,C) on the (family of) compact complex manifolds obtained by quotienting C^2-(0,O) by a given homothety. These manifolds are certainly not projective, as its first Betti number is 1.