Order of the subgroup of traceless matrices

Question

Let $M_n(\mathbb{F}_q)$ denote the ring of $n\times n$ matrices with entries in the finite field of $q$ elements, for prime $q$.

1. How many elements of $M_n(\mathbb{F}_q)$ have zero trace? and
2. Since every trace zero element of $M_n(\mathbb{F}_q)$ can be written as a commutator of two matrices, am I right in thinking that $\#\{x\in M_n(\mathbb{F}_q):Tr(x) = 0\} = |[M_n(\mathbb{F}_q),M_n(\mathbb{F}_q)]|$, where $[\cdot,\cdot]$ denotes the additive commutator and $[M_n(\mathbb{F}_q),M_n(\mathbb{F}_q)]$ is the subgroup of $M_n(\mathbb{F}_q)$ additively generated by all commutators? If not,
3. What is $|[M_n(\mathbb{F}_q),M_n(\mathbb{F}_q)]|$?