Yes, the projectivized orbit of a highest weight vector is well known to be the only closed one. It's also an orbit under the compact real form $G_u$, and as such is the unique $G_u$-orbit that is a complex (hence Kähler) submanifold (work of Borel, Weil, Tits, Hirzebruch). In this paper Kostant and Sternberg work out the conditions under which $G_u\cdot[v]$ is a symplectic submanifold. For an exposition see e.g. Guillemin and Sternberg's Symplectic Techniques in Physics.
(Orbits under other real forms are thoroughly studied in J. A. Wolf, The action of a real semisimple group on a complex flag manifold.)
Your problem, however, is that of studying the orbits in the flag manifold $\Bbb P(V)=GL(V)/GL(V)_{[v]}$ under any complex (irreducibly represented, reductive) $G\hookrightarrow GL(V)$. In this generality I'm not sure there is much one can say. For example, a theorem of Chevalley (J. Humphreys, Linear algebraic groups, 11.2) asserts that every $G/H$, $H$ Zariski closed in $G$, occurs as such an orbit for some (not necessarily irreducible) representation.