Work over an algebraically closed field ($\mathbb C$, if you prefer) and fix $g\geq 2$. By $M_g$ I mean, of course, the moduli space of smooth projective curves of genus $g$. I know it is in general not proper.
Question 1. Is it non proper for every $g$? And if not, what is an example of a proper $M_g$?
The motivation for the above question is an argument I read on some lecture notes. This is how I understood it: let us consider the (injective) Torelli morphism $M_g\to A_g$ (defined by $[C]\mapsto [J(C)]$), where $A_g$ is the moduli space of PPAVs. Then, if $M_g$ were proper, its image would coincide with the Torelli locus $T_g\subset A_g$. But $T_g$ contains products of PPAVs, and no such product can be the Jacobian of a curve. Contradiction.
Now I'm lost.
Question 2,3. How to see that $T_g$ contains products of PPAVs? And why a product is not the Jacobian of any curve?
Also, in passing, where can I find a clean definition of the theta divisor on a PPAV? I am confused about this point.
Thanks in advance.