2-Torsion in Jacobians of Curves Over Finite Fields

Question

Let $C$ be a (smooth, projective) curve over a finite field $\mathbb{F}_q$, and let $J_C(\mathbb{F}_q)$ denote its Jacobian. Suppose the genus $g$ of $C$ is at least $1$.

Question 1: Are there curves $C$ for which $J_C(\mathbb{F}_q)$ is isomorphic, as a group, to $(\mathbb{Z}/2\mathbb{Z})^k$ for some $k$? Can one characterize all such curves?

Question 2: What is known about upper bounds of the size of the $2$-torsion of $J_C(\mathbb{F}_q)$ compared to the size of $J_C(\mathbb{F}_q)$ itself? Is it necessarily true, say, that the 2-torsion is negligible if some parameter ($q$ or $g$) is large enough?

Answer 1

I think $y^2=x^9-x$ over $\mathbb{F}_3$ has $J_C(\mathbb{F}_3)$ isomorphic to $(\mathbb{Z}/2)^6$ but please check.

The $2$-torsion in $J_C$ over the algebraic closure is $(\mathbb{Z}/2)^{2g}$ (or smaller in characteristic two). On the other hand, $\#J_C(\mathbb{F}_q) \ge (\sqrt{q} -1)^{2g}$, so for $q > 9$, the latter is bigger than the former (and usually much bigger). So the things you want can only exist for small $q$. I don't know a classification.