Fernando Rodriguez Villegas has a neat little book "Experimental number theory", which has several sections on Galois groups. I am away from home so I can't check right now whether there is anything in there that's relevant, but there may well be.

As far as I know, there is still no a priori argument known which would prove that a given projective simply-connected Calabi-Yau threefold X contains a rational curve (let alone a balanced one).

Whenever the ample cone is finite rational polyhedral, adding all the primitive integer generators of the cone gives you an ample class which is invariant under automorphisms of your variety. Thus the whole automorphism group embeds in a projective linear group, so e.g. cannot be infinite discrete.