I suspect that the answer to question 2 should be negative thanks to Kamawata-Morrison conjecture about nef cones of Calabi-Yau varieties that is proven for surfaces (a recent reference is here http://arxiv.org/abs/1008.3825).
Indeed, it is known that on a minimal two-dimesnional surface of Kodaira dimesnion 0 the fundamental domain of the action of automorphism of the surface on the nef cone is rational polyhedral. It is sufficient to study ample classes that belong to one rational polyhedral domain. If it were true that every nef integral class on each Enriques surface is effective, then the answer to your question 2 should be indeed negative. This just should follow from the fact, the semi-group of integer vectors in every rational polyhedral cone is finitely generated, so only finitely many integer vectors can not be presented as a sum of two others.
PS I forgot that the canonical class of Enriques surface is not effective :) , though of course it is nef. Still, it could be that the above "argument" can be adjusted.