Let $M$ be a smooth manifold with a conformal structure $\mathcal{C}$, i.e. an equivalence class of metrics (say Lorentzian or Riemannian).
I read that the conformal structure $\mathcal{C}$ is determined by its null cones, i.e. the set of vectors $v\in T_pM\otimes\mathbb{C}$ such that $g_p(v,v)=0$ for some (and hence any) representative metric $g_p\in \mathcal{C}_p$.
Clearly if the conformal structure is known, then the null cones can be determined.
Question 1: How is the conformal structure determined when the null cones are known?
I think saying that a vector $v\in T_pM\otimes \mathbb{C}$ is null means saying $v$ is in the kernel of some quadratic form $Q_p$ on the tangent space. Is the conformal structure $\mathcal{C}_p$ then defined by the bilinear form $B_p$ associated to $Q_p$?
Question 2: How is the signature of the conformal class determined, i.e. if it is Riemannian, Lorentzian, or some other signature?
Question 3: How do we make sure the conformal structure varies smoothly from point to point, just by knowing its null cones?
Question 4: Why do we need to complexify the tangent space for this construction (or is this only necessary in special cases)?