There is one thing that I do not understand... They must have known that we would most likely not go another day without choosing someone to execute. It would have been their only guaranteed chance of converting someone... why choose a dog?

Your explanation makes sense if we assume that they converted Missile, but it does not explain why they chose him out of everyone else to convert. Is it not more likely that Missile merely did not know how to communicate with us right away, and when he tried to tell us that Levi was not the detective, we all assumed that he was a VIP and executed him?