This will never happen. The divisors $D$ on $C$ of degree $g+1$ such that $|D|$ has a base point are of the form $g^1_2+F$, with $F$ effective of degree $g-1$; in $J^{g+1}C$, they form a divisor $\Delta $ which is a copy of the theta divisor. For any $E\neq 0$ in $JC$ (2-torsion or not), the translate $\Delta -E$ is different from $\Delta $, therefore there exists $D\in (\Delta -E)\smallsetminus \Delta $; then $D$ is base-point free, but $D+E$ belongs to $\Delta $, hence has a base point.

This will never happen. The linear systems of degree $g+1$ on $C$ with a base point are of the form $g^1_2+F$, with $F$ effective of degree $g-1$; in $J^{g+1}C$, they form a divisor $\Delta $ which is a copy of the theta divisor. For any $E\neq 0$ in $JC$ (2-torsion or not), the translate $\Delta -E$ is different from $\Delta $, therefore there exists $D\in (\Delta -E)\smallsetminus \Delta $; then $|D|$ is base-point free, but $|D+E|$ belongs to $\Delta $, hence has a base point.