This is regarding question 1. There is a much earlier approach through Atiyah-Singer G-signature theorem that works for certain divisible classes. 

So if $g$ is the genus of an embedded surface $S$ representing $a$ then (under certain assumption) one gets the following inequality
$$
\beta_2+2g\ge|\frac{1}{2}a\cdot a-\sigma|
$$
where $\sigma$ is the signature of the intersection form on the second homology and $\beta_2$ is the second Betti number. (I am not sure about the coefficient 1/2 on the right, it is more complicated actually, but I think morally it's ok.)

This inequality applies to even classes $2nx$ in $\mathbb{CP}^2$ and $2nx+2my$ in $S^2\times S^2$
resulting in
$$
g\ge n^2-1,\;\;\; and\;\; g\ge 2nm-1
$$
respectively. (Also it applies to other divisible classes.)

One has to find a finite cyclic group of order $k$, $k|a$ acting on $M$ so that the fixed point set is a surface $X$. Then look at a $k$-fold cover $\tilde M$ that branches over $X$ and apply the G-signature theorem to this cover.

I think the proof can be found in Rohlin's "Two dimensional sub manifolds of 4 dimensional manifolds" or in Hsiang, Szczarba, "On embedding surfaces in four-manifolds".