On any compact complex manifold, the set of all global holomorphic vector fields is a finite dimensional Lie algebra. They span the tangent space at every point just when each component of the manifold is a homogeneous for the action of its biholomorphism group. Read the book by Dmitri Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, vol. E27, Friedr. Vieweg, Braunschweig and Wiesbaden, 1995, vii + 201 pp., $49.00, ISBN 3-528-06420-X.
If $c_1 \le 0$, then they are tori, by Bochner, I believe. Check Wang's paper on homogeneous complex manifolds.
They are strongly dominable, since the Lie group of biholomorphisms has an exponential map.