Question 1: Let $X$ be a smooth rational surface with anti-canonical cycle, i.e, $-K_X$ is effective and its irreducible components form a polygon. Say, assume that $-K_X=\sum_\limits{i=1}^N D_i$ and $D_i^2=a_i$ (self intersection numbers).I wonder whether there exists a criterion for a divisor,say $N$ being nef on this surface. It seems that if I want to show that $-K_X$ is nef, It is enough to show that $-K_X.D_i=D_i^2+2\geq 0$, i.e:$a_i\geq -2$ for each $i$, I wonder whether there exists similar criterion for general divisor $D$ without computing intersection numbers for each irreducible curves on $X$ but with certain class(with finitely many members) of irreducible class of curves on $X$. The example I have in my mind is a toric surface $X$. Since all the divisors are linear equivalent to the boundary divisors, so one just need to verify the Criterion on boundary divisors.
Note that $(X,-K_X=D)$ is called an anti canonical pair or Looijenga pair. It seems that there is a statement saying that any Looijenga pair has a toric model. I wonder whether we can obtain some "convenient" criterion from this connections.
Question 2: Let $X$ be a rational surface with $K_X^2>0$, i.e $X$ is a rational surface with big anti canonical class. I wonder whether in this setting, we have "convenient" criterion for a divisor being nef. Note that one is able to show that $X$ is actually a Mori-dream space(they called rational Mori dream surface).And Mori-dream space is natural generalization of toric varieties. I wonder whether this helps to establish some convenient criterion.
Question 3: Since those two type of surfaces are "close" to toric surface in some sense,I wonder whether algebraic geometry on them also have some "combinatoric" flavor using convex cones,dual cones,etc.
Thanks
Maybe this can answer part of your questions. (In any dimensions)
Consider $\overline{NE}(X)\subset N_1(X)$ the closure of cone generated by effective $1$-cycles. Then the nef cone of divisors is dual to $\overline{NE}(X)$. That is, to test nefness, theoretically you need to test all curves lying in $\overline{NE}(X)$, but actually you only need to test all curves "generating" $\overline{NE}(X)$ as a cone.
In Mori Dream space case (including toric case), $\overline{NE}(X)$ is generated by finitely many extremal rays, and each extremal ray is generated by a curve. Hence there should exists only finitely many curves for which nefness can be tested on. If $\overline{NE}(X)$ is not as good as above, then one might not expect there exist finitely many "test curves".