It was a hard to figure out episode.
Maybe the Silence are hanging around with Spaceship girl, to make sure people don't know about her?
Took me a minute to realize why The Doctor went with the Moon Landing Video. I originally thought that the subliminal message would only work on the people alive to watch it live. But that wouldn't really work, once all those people pass away from old age. But then it hit me, they put the message on the signal right on the moon, and that anytime, anyone watches it, they get a "recharged command" to kill The Silence on sight. And it would make sense that it would be a moment watched for the rest of human history, the command would too, be watched for the rest of human history.
I imagine the storyline of what bother with the doctor being in Area 51 and the roundup of the companions, was a coverup, to make sure the Silence would be caught off guard, thinking they were in this inescapable box. The Companion's went out to see how far the Silence reach was, then come back in. With the Box built around the invisible TARDIS, they could slip out of there undetected.