This answer is rewritten and include more details

First of all I highly recommend you the article of Paul Biran *From Symplectic Packing to Algebraic Geometry and Back* available on the page  http://www.math.tau.ac.il/~biranp/Publications/Pubications.html , especially theorem 3.2.

Your question basically asks "*what is the symplectic cone of $\mathbb CP^2$ blown up in a finite number of points?*" This question was answered by Paul Biran (check theorem 3.2. from the above article), though the answer is not 100% explicit.
Also, it is known that the symplectic cone of $\mathbb CP^2$ blown up in up to $9$ points coincides with the Kahler cone if the points are chosen so that the resulting surface has only $-1$ curves (in particular it is Fano is the number of points is at most $8$). This permits to answer you last question (that is done below). In fact Kahler cones of Fano surfaces rather classical objects and all basic questions about them can be answered.

I would like to add that from a certain conjecture from algebraic geometry - *Harbourne-Hirschowitz* conjecture, it follows that the Kahler cone of $\mathbb CP^2$ blown up in a very generic collection of points  coincides with its symplectic cone (Literately this conjecture say the following 
: *any integral curve with negative self-intersection on the blow-up of $\mathbb CP^2$ 
at a set of points in very general position is a smooth rational curve with self-intersection $−1$*. In order to deduce the statement that symplectic cone coincides with the Kahler one you have to use SW theory). Habourine-Hirschwitz conjecture is open even for $\mathbb CP^2$ blown up in $10$ points, and the famous Nagata conjecture is a partial case of it.


Now let us answer the last bit of the question. **The class $2H-E_1-E_2$ is not symplectic**. 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.