If there were an impact on the Moon, would it maintain a stable orbit around the Earth?

Answer

The energy of the impact changes the lunar orbit. In principle, this already applies to every speck of dust that falls on the Moon, but of course the effect is small. To have a meaningful change of orbit, you need a very large impact, and the double question then is whether sufficiently large threatening objects exist, and if so, whether the Moon itself will survive that impact anyway!

The largest object that is reasonably realistic about impacting the Moon is a small planet ten miles in diameter, the kind that once killed the dinosaurs. Such an object has a mass of the order of a few hundred millionths of the Moon. If it collides with the Moon at a speed ten times greater than that of the Moon, you change the Moon’s speed by a few ten-millionths at most. The change in the distance from the Moon to the Earth is therefore of that order, a hundred meters at most.

The effect is instantaneous. At the impact, the Moon gets a new orbit, and that orbit is then immediately permanent, until a new impact may follow. The track is therefore stable. Unless the orbit brings the Moon so close to Earth that significant tidal forces play a role, or friction with the atmosphere, and then it ends quickly. But this won’t happen.

Furthermore: by definition an orbit ‘where the Moon goes out again’ is not a stable orbit!