Roots of $x^p + x + [\alpha]_p \in \mathbb{F}_p[x]$

As you note, by Euler we have $x^p\equiv x\pmod p$ for all $x\in\Bbb{Z}_p$, so $$x^p+x+[\alpha]_p\equiv 2x+[\alpha]_p,$$ holds for all $x\in\Bbb{Z}_p$. So if $p\neq2$ then $x=-\tfrac{1}{2}[\alpha]_p$ is a root, and it is in fact the only root.

Consider $$\gcd(x^p+x+\alpha,x^p-x)=\gcd(2x+\alpha,x^p-x),$$ where $\gcd$ denotes the greatest common divisor in $\mathbb Z_p$. Observe that, if the degree of the gcdis positive, then it is equal to the number of distinct roots of $g(x)$ in $\mathbb Z_p$.

We have two cases:

1. If $p\neq 2$ then $$\gcd(2x+\alpha,x^p-x)=x+2^{-1}\alpha,$$ and $-2^{-1}\alpha$ is the unique root in $\mathbb Z_p$, for every $\alpha$.
2. If $p=2$, then you already have the solution.