Edit: As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: assuming $p\leq q$, the maximum number $\ell$ of points of $C$ lying on a line is $\leq q\ $ — otherwise the line is contained in $C$. By B. Basili, Indice de Clifford des intersections complètes de l’espace, Bull. S. M. F. 124, no. 1 (1996), p. 61-95, the gonality of $C$ is $pq-\ell$, hence $\geq p(q-1)$. But if $C$ is a 2-sheeted cover of a curve $D$ of genus $\leq 3$, a $g^{1}_3$ on $D$ lifts to a $g^1_6$ on $C$; similarly if $C$ is a 3-sheeted cover of an elliptic (or more generally hyperelliptic) curve $E$, a $g^1_2$ on $E$ lifts to a $g^1_6$ on $C$. Hence the gonality of $C$ is $\leq 6$, which is less than $p(q-1)$.

Edit: As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: suppose $C$ is a $(p,q)$ complete intersection, with $p\geq q$. The maximum number $\ell$ of points of $C$ lying on a line is $\leq p\ $ — otherwise the line is contained in $C$. By B. Basili, Indice de Clifford des intersections complètes de l’espace, Bull. S. M. F. 124, no. 1 (1996), p. 61-95, the gonality of $C$ is $pq-\ell$, hence $\geq p(q-1)$. But if $C$ is a 2-sheeted cover of a curve $D$ of genus $\leq 3$, a $g^{1}_3$ on $D$ lifts to a $g^1_6$ on $C$; similarly if $C$ is a 3-sheeted cover of an elliptic (or more generally hyperelliptic) curve $E$, a $g^1_2$ on $E$ lifts to a $g^1_6$ on $C$. Hence the gonality of $C$ is $\leq 6$, which is less than $p(q-1)$ if $q\geq 4$.

No, this cannot happen. Suppose $C$ is a $(p,q)$ complete intersection. The line bundle $\mathscr{O}_C(1)$ induced by $\mathscr{O}_{\mathbb{P}^3}(1)$ is the only line bundle $L$ on $C$ with $L^{p+q-4}=K_C$ and $h^0(L)=4$; therefore any automorphism $u$ of $C$ comes from an automorphism $\tilde{u} $ of $\mathbb{P}^3$. The fixed locus of $\tilde{u} $ is a union of linear subspaces; it follows easily that $u$ has at most $pq+1$ fixed points. Then the Riemann-Hurwitz formula implies that the genus $g$ of the quotient curve is large. For instance, for $p=q=4$, you get $g\geq 13$ if $u^2=\operatorname{Id} $, and $g\geq 9$ if $u^3=\operatorname{Id} $.

Edit: As pointed out by Sasha, my original argument was not complete for case (2). One can argue as follows: assuming $p\leq q$, the maximum number $\ell$ of points of $C$ lying on a line is $\leq q\ $ — otherwise the line is contained in $C$. By B. Basili, Indice de Clifford des intersections complètes de l’espace, Bull. S. M. F. 124, no. 1 (1996), p. 61-95, the gonality of $C$ is $pq-\ell$, hence $\geq p(q-1)$. But if $C$ is a 2-sheeted cover of a curve $D$ of genus $\leq 3$, a $g^{1}_3$ on $D$ lifts to a $g^1_6$ on $C$; similarly if $C$ is a 3-sheeted cover of an elliptic (or more generally hyperelliptic) curve $E$, a $g^1_2$ on $E$ lifts to a $g^1_6$ on $C$. Hence the gonality of $C$ is $\leq 6$, which is less than $p(q-1)$.