What would happen to the Earth's momentum theoretically speaking if the moon was destroyed by some outside body such as a asteroid ?
depends on the details ... basically the asteroids momentum would get added to the system.
Not in a significant amount for reasonable masses in spaceflight.Would landing on the Moon and leaving the lander behind (Mass) have an affect on this ?
According to wikipedia, the total mass of all man-made objects on the moon is about 180 tons.@mfb
Describe "not a significant amount" after 43 years + however much more junk is up there from all the other probes/landers, please.