computational physics - Coordinate system for numerical simulation of general relativity

Lets say i want to simulate the differential equations of GR with some numerical method. I can express the Einstein tensor in terms of the christoffel symbols which in turn can be expressed in terms of the metric and its first and second derivative.

Now i can impose a set of coordinates $[t, x, y, z]$ and set up a big cartesian grid. Each point contains information about the metric of that point, as seen by an observer at infinity. Initially the space is empty so the metric will reduce to the Minkowski-metric.

Now i place some mass at the origin, but with finite density. Assume that the mass encapsulates many grind points and the grid extends to a far distance so the metric at the end of the grid is approximatly flat.

Now i want to simulate what happens. To do this i rearrange the equations and solve for $\frac{\partial^2}{\partial t^2}g_{\mu\nu}$ which should govern the evolution of the system. Since i know the initial conditions of the metric and $T_{\mu\nu}$ i should be able to simulate the dynamics of what happens.

(I assume the metric outside would converge to the outer schwarzschild metric while the parts inside the mass would converge to the inner schwarzschild metric. Additionally a gravitational wave should radiate away because of the sudden appearance of a mass).

However, by doing so i have placed the spacetime itself on a background grid, which seems fishy to me.

Question 1: How does the coordinate system influences the equation? For example i could have choosen $[t, r, \phi, \theta]$ and would have gotten the same equations since it involves only ordinary derivatives. Do i assume correctly that the coordinate system properties only appear during numerical integration?

Question 2: What physical significance does this "cartesian grid" system have? If i look at a point near the surface of the mass after a long time, where is this point in the spacetime? A stationary observer would follow the curvature and may already have fallen into the mass. Does this mean my coordinate system itself "moves" along? How can i get a "stationary" (constant proper radial distance) coordinate system?

Question 3: Since i have the metric at every grid point i could calculate (numerically) the geodesic through this spacetime and find the path of an infalling observer, right?

Question 4: Would this work for simulating non-singular spacetimes? Or is this approach too easy?

edit1: To clarify question 1, different coordinate systems have different artefacts on their own. Instead of a grid i could use schwarzschild-coordinates. If i now expand the whole equation i would get the same form, because it is coordinate independent. However the resulting metric would look different (same metric, but expressed in another coordinate system). I'm a bit confused because the metric and the coordinate system are tied together. If i'm solving for the metric itself i still need to provide the coordinate system. And i dont understand how i introduce the coordinate system.