Since the signature of a Lorentzian manifold (M,g) is (-,+,+,+), am I right to assume that the determinant of $g_{ij}$ with EACH coordinate system is $-1$?
Metric on Lorentzian manifold
differential-geometrymetric-spacessemi-riemannian-geometry
Related Solutions
Let $v_1, \dots, v_n$ be a basis for $T_pM$ such that $g(p)(v_i, v_i) > 0$ for $i = 1, \dots, k$, and $g(p)(v_i, v_i) < 0$ for $i = k + 1, \dots, n$. We can extend this basis to a basis of local smooth vector fields $V_1, \dots, V_n$ for $TM|_U$ where $U$ is an open neighbourhood of $p$.
Now consider the functions $f_i : U \to \mathbb{R}$ given by $f_i(q) = g(q)(V_i|_q, V_i|_q)$. Note that $f_i$ is smooth and $f_i(p) = g(p)(V_i|_p, V_i|_p) = g(p)(v_i, v_i)$ which has a sign. For $i = 1, \dots, k$, let $U_i = f_i^{-1}((0, \infty))$ and for $i = k + 1, \dots, n$ let $U_i = f_i^{-1}((-\infty, 0))$. Then on $U' = U_1\cap\dots\cap U_k\cap U_{k+1}\cap\dots\cap U_n$ each $f_i$ has a sign. For any $q \in U'$, $f_i(q) > 0$ for $i = 1, \dots, k$ and $f_i(q) < 0$ for $i = k + 1, \dots, n$, so $g(q)$ has the same index of $g(p)$. That is, the index of $g$ is a locally constant integer-valued function and is therefore constant on each connected component of $M$.
The definition of a Cauchy hypersurface you cited only implies that it is a continuously ($C^0$) embedded submanifold. This leads to $M$ being only homeomorphic to $\mathbb{R}\times S$. That the definition of global hyperbolicity (your item 4. is the actual original definition) is equivalent to the existence of a ($C^0$) Cauchy hypersurface and the topological splitting $M=\mathbb{R}\times S$ is the classical result proven by Geroch in the 1970s.
On your questions: The Cauchy hypersurface is smooth if it is a smoothly embedded submanifold (usually $C^\infty$). This is an additional condition for the Cauchy hypersurface. The Cauchy hypersurface is spacelike if additionally the restriction of the metric is Riemannian or, equivalently, all inextendible causal curves hit it at most once. This is another additional condition put on the Cauchy hypersurface.
For a long time it was an open question if a globally hyperbolic spacetime actually always admits a smooth spacelike hypersurface and hence if it is actually diffeomorphic to $\mathbb{R}\times S$. This was only proven at the begin of this century by Bernal and Sánchez. You might want to consult their papers for details:
Best Answer
Yes, you are.
If you change the coordinate system the metric in the new coordinates will be the conjugate of $g_{ij}$ by the coordinate change’s differential map. Since the differential belongs to the general linear group, this will not change neither the metric’s signature nor eigenvalues. Thus, the determinant shall remain the same.