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.

The answer to question 2 is 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), and to the comment of Damiano below.

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 (see the above reference). So it is sufficient to consider only ample classes that belong to one rational polyhedral fundamental domain. Notice that every nef integral class (apart from $K$) on each Enriques surface is effective (see the proof of Damiano). Now notice that since the fundamental domain is rational polyhedral, the semi-group of integer vectors in it is finitely generated, so only finitely many integer vectors can not be presented as a sum of two others. In other words, only finite number of ample divisors in a given fundamental domain are not linearly equivalent to a sum of several divisors. Since by definition all domains are the same, the statement is proven.

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.

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.