Can the Jacobian Be 0?

If the Jacobian is zero, it means that there is no change whatsoever, and this means you get an overall change of zero at that point (with respect to the rate of change with respect to the expansion and contraction with respect to the entire volume).

Does Jacobian have to be positive?

Please remember that the Jacobian defined here is always positive. Exercises: 24.2 What is the relationship between the Jacobian going from dxdy to dsdt, and that going the opposite way?

Is the Jacobian a constant?

For your first question, a constant Jacobian does not necessarily mean that the function is linear. For your second question, you don't need the Jacobian to be constant, you just need it to be non-zero.