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 $$ \beta_2+2g\ge\left|\frac{1}{2}a\cdot a-\sigma(M)\right| $$ where $\sigma(M)$ 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.)

Assume $a=kb$, where $k$ is an integer. One has to look at $k$-fold cover $\tilde M$ that branches over $S$ and apply the G-signature theorem to this cover. This gives a formula for $\sigma(\tilde M)$ $$ \sigma(\tilde M)=k\sigma(M)-\frac{(k^2-1)a\cdot a}{3k}. $$ Then the final estimate comes from comparing Euler characteristic, signatures and second Betti numbers.

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".

It is interesting that Rohlin remarks that first he obtained the above display formula without using G-signature formula from a "corresponding version of cobordism theory."

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 assumptions) one gets the following inequality $$ \beta_2+2g\ge\left|\frac{1}{2}a\cdot a-\sigma(M)\right| $$ where $\sigma(M)$ 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.)

Assume $a=kb$, where $k$ is an integer. One has to look at $k$-fold cover $\tilde M$ that branches over $S$ and apply the G-signature theorem to this cover. This gives a formula for $\sigma(\tilde M)$ $$ \sigma(\tilde M)=k\sigma(M)-\frac{(k^2-1)a\cdot a}{3k}. $$ Then the final estimate comes from comparing Euler characteristics, signatures and second Betti numbers.

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

It is interesting that Rohlin remarks that he had the above display formula from "a corresponding version of cobordism theory" without using Atiyah-Singer.

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".

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 $$ \beta_2+2g\ge\left|\frac{1}{2}a\cdot a-\sigma(M)\right| $$ where $\sigma(M)$ 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.)

Assume $a=kb$, where $k$ is an integer. One has to look at $k$-fold cover $\tilde M$ that branches over $S$ and apply the G-signature theorem to this cover. This gives a formula for $\sigma(\tilde M)$ $$ \sigma(\tilde M)=k\sigma(M)-\frac{(k^2-1)a\cdot a}{3k}. $$ Then the final estimate comes from comparing Euler characteristic, signatures and second Betti numbers.

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".

It is interesting that Rohlin remarks that first he obtained the above display formula without using G-signature formula from a "corresponding version of cobordism theory."