For a pseudo-Riemannian manifold M of signature $(p,q)$, would it be valid to say that every frame (ordered basis) $e\in LM$ contains $p$ time-like vectors and $q$ space-like vectors?
If this does not hold, is this true for the weaker case of a Lorenzian manifold?
No. Consider the usual Minkowski metric with $\eta = \text{diag}(1,-1,-1,-1)$. Then $(1,0,0,0), (1,1,0,0), (1,1,1,0), (1,1,1,1)$ is an ordered basis with 1 time-like vector, 1 null vector, and 2 space-like vectors.