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.