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.