general relativity - Calculating the determinant of a metric tensor

Suppose the line element is

$$ds^2 = -A(t,r)^2dt^2+B^2(t,r)dr^2+C^2(t,r)d\theta^2+C^2\sin^2\theta d\phi^2.$$
Since the metric is diagonal, to find the determinant I can multiply the diagonal entries, $$\det g_{ab} = g = -A^2B^2C^4 \sin^2\theta.$$ I have a few questions about this.

1. First off, why do we call the metric determinant $g$?



2. Why isn't it true that $g = g_{ab} g^{ab} = 4$? Isn't that how $g$ is defined?


3. When will it be true that $g = 1$?