Dear Fellow,
You can't move $E$ (!), hence there is no contradiction with it having self-intersection -1.
Indeed, if you take a normal vector field along $E$, it will necessarily have degree -1 (i.e.
the total number of poles is one more than the total number of zeroes), or (equivalently),
the normal bundle to $E$ in the blown-up surface is $\mathcal O(-1)$.
[Added:] Here is a version of the argument given in David Speyer's answer, which is rigorous modulo basic facts about intersection theory:
Choose two smooth very ample curves $C_1$ and $C_2$ passing through the point $P$ being blown-up in different tangent directions. (We can construct these using hyperplane sections in some projective embedding, using Bertini; smoothness is just because I want $P$ to be a simple point on each of them.) If the $C_i$ meet in $n$ points away from $P$, then
$C_1\cdot C_2 = n+1$.
Now pull-back the $C_i$ to curves $D_i$ in the blow-up. We have
$D_1 \cdot D_2 = n + 1$. Now because $C_i$ passes through $P$, each $D_i$ has the form
$D_i = D_i' + E,$ where $D_i'$ is the proper transform of $C_i$, and passes through $E$ in a single point (corresponding to the tangent direction along which $C_i$ passed through $P$).
Thus $D_1'\cdot D_2' = n$ (away from $P$, nothing has changed, but at $P$, we have separated
the curves $C_1$ and $C_2$ via our blow-up).
Now compute $n+1 = D_1\cdot D_2 = D_1'\cdot D_2' + D_1'\cdot E + E\cdot D_2' + E\cdot E
= n + 1 + 1 +
E\cdot E$, showing that $E\cdot E = -1$. (As is often done, we compute the intersection of curves that we can't move into a proper intersection by adding enough extra stuff that we can compute the resulting intersection by moving the curves into proper position.)
A more direct approach is the following:
Let $X$ be the projective bundle $\pi:\mathbb{P}(\mathcal{E})\to \mathbb{P}^1$ where $\mathcal{E}=\mathcal{O}\oplus \mathcal{O}(-1) \oplus \mathcal{O}(-2)$. Let $M$ be the tautological bundle of $X$. It is easily checked that the ample line divisors $H_i$ on $X$ correspond to line bundles of the form $M^a\otimes \pi^*\mathcal{O}(b)$ with $b>2a$. We show that $Mov(X)$ is not spanned by products of the form $H_1\cdot H_2$.
Consider the line bundle $L=M\otimes \pi^*\mathcal{O}(-1)$. Using the Leray spectral sequence for the morphism $\pi$ we easily see that $L$ is not pseudoeffective. However, it is also straightforward to check that $$L\cdot H_1\cdot H_2=b_1+b_2-4>0$$ for $H_i=M\otimes \pi^*\mathcal{O}(b_i)$. Hence $L$ lies in the dual cone of $\overline{Q}(X)$ (using your notation). Now, if $\overline{Mov}(X)$ was generated by the $H_1\cdot H_2$'s this would imply that $L$ is pseudoeffective (by BDPP), a contradiction.
Best Answer
Yes, this is the standard example of a variety whose cone of curves has infinitely many extremal rays. (A reference is Koll\'ar--Mori, p.22).
To see why there are infinitely many (-1)-curves: each of the 9 points you blow up gives a (-1)-curve E_i, as you know. But the E_i are also sections of the elliptic fibration determined by the pencil. So one can use the group structure of the generic fibre (which is an elliptic curve over the function field k(P^1)) to translate any of the E_i to any other section of the fibration; since the action is by automorphisms of the surface, the other sections must be (-1)-curves also. As long as there are infinitely many sections (which is guaranteed by the assumption that the cubics are general) one gets infinitely many (-1)-curves this way.
Edit: It's worth mentioning that there are other examples of surfaces with non-finitely generated cone of curves, which differ from the example above in an interesting way. For example, suppose X is a K3 surface which has Picard rank at least 3 and no (-2) curves. Then the (closed) cone of curves of X consists of all classes in N^1(X) satisfying x^2 ≥ 0 and x.H ≥ 0 for any fixed ample class H. This is the standard "round cone" in N^1, with uncountably many extremal rays. (A similar example is obtained by taking X to be an abelian surface with Picard rank at least 3.)
As mentioned in the comments, Volume 1 of Lazarsfeld's great book Positivity in Algebraic Geometry is the best reference for these kinds of questions.