By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.

Namely, if a surface $S$ is uniruled, all plurigenera vanish, and that's all you need. Indeed, if $q=0$, the result follows by Castelnuovo rationality criterion. For the irregular ($q>0$) case, Beuville shows, (Proposition VI.15.(1)) that, if the surface is not ruled, then either $P_4 \neq 0$ or $P_6 \neq 0$.

Of course all the mentioned results are highly non trivial, but all proofs are in Beauville's book.

By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.

Namely, if $S$ is uniruled, then you can solve the indeterminacy (Theorem II.7) and get a dominant morphism $X \rightarrow S$ where $X$ is birational to a product $C \times {\mathbb P}^1$. It follows, by Proposition III.20, that (being birational to a surface with vanishing plurigenera) all plurigenera of $X$ vanish. The argument of the proof of the same proposition applied to the dominant map $X \rightarrow S$ show that $P_n(S) \leq P_n(X)$ and therefore all plurigenera of $S$ vanish too.

Then, if $q=0$, the result follows by Castelnuovo rationality criterion. For the irregular ($q>0$) case, Beuville shows, (Proposition VI.15.(1)) that, if the surface is not ruled, then either $P_4 \neq 0$ or $P_6 \neq 0$.

Of course all the mentioned results are highly non trivial, but all proofs are in Beauville's book.

By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.

Namely, if a surface $S$ is uniruled, all plurigenera vanish, and that's all you need. Indeed, if $q=0$, the surface is rational directly by Castelnuovo rationality criterion. For the irregular $q>0$ case, you can use Chapter VI of Beauville's book, devoted to minimal surfaces with $p_g=0$ and $q>1$: he shows, among other things (proposition VI.15.(1)) that if the surface is non-ruled, than either $P_4 \neq 0$ or $P_6 \neq 0$.

Of course all the mentioned results are highly non trivial, but all proofs are in Beauville's book.

By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.

Namely, if a surface $S$ is uniruled, all plurigenera vanish, and that's all you need. Indeed, if $q=0$, the result follows by Castelnuovo rationality criterion. For the irregular ($q>0$) case, Beuville shows, (Proposition VI.15.(1)) that, if the surface is not ruled, then either $P_4 \neq 0$ or $P_6 \neq 0$.

Of course all the mentioned results are highly non trivial, but all proofs are in Beauville's book.