Finite/Infinite groups

(a) For $n\geq 1$, $\prod_{i=1}^n\Bbb{Z}_2$ is a finite group whose every nontrivial element has order $2$. So there are infinitely many such groups.

(b) $\prod_{i=1}^{\infty}\Bbb{Z}_2$ is the desired group.

(c) You may consider the example $\Bbb{Q}/\Bbb{Z}$. Then for every $n\in \Bbb{N}$, $\frac{1}{n}+\Bbb{Z}$ is element of order $n$ in $\Bbb{Q}/\Bbb{Z}$.

(d) For $n\geq 2$, $\prod_{i=1}^{\infty}\Bbb{Z}_n$ is an infinite group whose every nontrivial element has finite order. So there are infinitely many such groups.