The dynamic metric is the metric for the quasidiagonal system; and so for self consistency the solution can only be proven to exist (and be unique) when the metric (i.e. the solution itself) is hyperbolic/Lorentzian.
Noncompactness has zero impact whatsoever. Hyperbolic equations have finite speed of propagation and hence "local local uniqueness" (one local for in time, and one local for in space).
Finally, about $T$: time functions in Relativity theory is not god given. With diffeomorphism invariance you can always re-coordinate an open spacetime neighborhood of the spacelike hypersurface $M$ as $[0,T]\times M$. On the other hand, there are times where you can prove that on your coordinate system $|g(\partial_t, \partial_t)|$ has a lower bound; in this case usually what happened is that you assumed enough decay on the initial data (in some weighted Sobolev space, say) that you can in fact prove using energy estimates that you have good global foliations. In situations like this you automatically also get uniform boundeness on, say, the connection coefficients, and then your continuity argument will have no problem.
