In atmosphere, yes. In vacuum, like on the moon, no. The reason you feel force on Earth is because molecules from the engine bump into adjacent air molecules, which bump into molecules next to it, and so on until it hits your skin. It's called continuum flow. In vacuum, you have what's called free molecular flow, that is, individual molecules are spaced far apart relative to each other, and their effects are isolated. So, in fact, you would be hard press to see any effect on the surface of the moon.

What I don't understand is, wouldn't you think that if the government were trying to fool you, they would have thought of all these things you're thinking of right now? The things you are pointing out are not subtle, they are very obvious. I mean, the government has billions of dollars at its disposal and plenty of smart people working for it (especially in defense, intelligence, and at NASA, despite what the public may think). In fact, I would be more likely to believe the moon landings to be a hoax if the photos showed that the surface of the Moon behaves like the surface of the Earth, because it doesn't.