Hi YCho, I don't quite get why you say that your question is stupid...

Let me first answer the last bit. The class $2H-E_1-E_2$ is not symplectic. The point is that the symplectic cone of $\mathbb CP^2$ blown up in $2$ points coincides with the Kahler cone of $\mathbb CP^2$ blown up in two points. And $H-E_1-E_2$ is an exceptional $-1$-curve on this surface, while $(2H-E_1-E_2)\cdot (H-E_1-E_2)=0$, so $2H-E_1-E_2$ is not ample, and hence not symplectic. 

In general it is known that the symplectic cone of $\mathbb CP^2$ blown up in up to $9$ points coincides with the Kahler cone of the corresponding Delpezzo surface (if the number of points is $<9$ ), and the later cone is a classical enough object (if the number of points is at most $8$), so you would be able to find its description. There is a theorem of Biran that gives an answer to your question in general, but it is not completely explicit, you can check:

http://www.math.tau.ac.il/~biranp/Publications/Stbl_Pack.pdf