What might happen to the lunar atmosphere after a time of major human presence or activity on Earth's natural satellite? After hundreds of touchdowns and flyoffs and disturbances to environment?
The Moon does have an atmosphere. It is about 1,000,000 particles per m2, this is not very much compared to Earths 10,000,000,000,000,000,000. It is about comparable to the atmosphere that exists in the same region that the ISS orbits - 330 - 435km.
If you launched a rocket whatever gases were produced would likely not hang around long, of course there would be trace amounts and exactly what gases would depend on the fuel/oxidizer used in the rocket.
The moon's atmosphere currently consists of helium, argon and possibly neon, ammonia, methane and carbon dioxide. This discovery was made by the Lunar Atmospheric Composition Experiment (LACE) that was placed on the lunar surface on the Apollo 17 mission. On Earth we have used telescopes to detect sodium and potassium in the atmosphere.