$x^p-x+a$ irreducible for nonzero $a\in K$ a field of characteristic $p$ prime

Let $K$ be any field of characteristic $p$ with more than $p$ elements. Then $x^p-x$ has only the $p$ elements of the prime field as roots. Pick $b$ not in the prime field and let $a=-(b^p-b)$. Then $x^p-x+a$ has $b$ as root (and $b+k$ for $k$ in the prime field).

No, in general it is not true that the Artin-Schreier polynomial $x^p-x+\alpha$ is irreducible in any field of characteristic $p$. For instance, if $K$ is algebraically closed, the polynomial is obviously never irreducible.

Or, just take $K=\mathbb Z_p[y]/\langle y^p-y+1\rangle$ for the most basic counterexample, since $y^p-y+1$ is prime in $\mathbb Z_p[y]$. Then But $x^p-x+1$ has a root $y$ in $K$.