As an alternative to Francesco's nice geometric argument, the answer to question 1 is no, by definition of "hyperelliptic". I.e. a curve is hypereliptic if it has a line bundle of degree 2 with more than one section. If g = 3, any effective even theta charcteristic is such a line bundle. As to the subvariety M(1,g), this was (mostly) understood already by Riemann. I.e. if we look at the image of M(g) in the space of prin.pol. abelian varieties, then M(1,g) is just the intersection of Jacobians with the ample divisor defined by the vanishing of the theta function, hence it has pure codimension one, and is non empty.

As an alternative to Francesco's nice geometric argument, the answer to question 1 is no, by definition of "hyperelliptic". I.e. a curve is hypereliptic if it has a line bundle of degree 2 with more than one section. If g = 3, any effective even theta charcteristic is such a line bundle. As to the subvariety M(1,g), this was (mostly) understood already by Riemann. I.e. if we look at the image of M(g) in the space of prin.pol. abelian varieties, then M(1,g) is just the intersection of Jacobians with the ample divisor defined by the vanishing of the theta function, hence it has pure codimension one, and is non empty. If g-1 is twice an odd number, the presence of hyperelliptics in this locus avoids the use of ampleness.