For the first question take local harmonic coordinates the equation takes the form

$$ \partial^2_{tt} g_{ij} - g^{kl} \partial^2_{kl} g_{ij} = l.o.t. $$

and so using finite speed of propagation local existence holds in the non-compact case provided that the data is not too wild near infinity.

In fact, for any sufficiently smooth initial data, regardless of asymptotic behaviour, standard hyperbolic PDE theory tells you that there exists a sufficiently smooth Cauchy development. The only issue is that what is the "time of existence", as generally for non-compact initial data you may not have a lower bound on the time of existence.

For now assume that your initial noncompact manifold is geodescially complete. In the incomplete case you will have problems coming from the boundary behaviour.

As Deane commented that it suffices that the metric and its first derivative decays in the initial data. But you can get away with even weaker statements. For example, I am pretty sure it suffices to have

- Uniform lower bound on the radius of injectivity of the initial metric, by rescaling we can take this bound to be 1.
- Uniform upper bound on the local higher energy: for some $N$ depending on the dimension of your manifold you require
$$ \sup_{x\in M} \sup_{0 \leq k \leq N} \| \nabla^k \mathrm{Riem} \|_{L^2(B(x,1))} < \infty $$
and something analogous for the initial velocity: let $h = \partial_t g | _{t = 0}$ be the symmetric two tensor representing the initial velocity, require
$$ \sup_{x \in M} \sup_{0 \leq k \leq N-1} \|\nabla^k h\|_{L^2(B(x,1))} < \infty $$

Then using finite speed of propagation and what you already know about in the compact case you get control on the time of existence of the Cauchy developments originating from balls of radius one. Alternatively you can get the same result by modifying one of the many write-ups of the well-known proof in the case of Einstein's equation.

I leave the second question to other experts.