The government doesn't have to stop its work in order to tell the public. In fact, given the lack of goa'uld spies on Earth, there's no reason for the information to be top secret. Telling the public doesn't force the politicians or the military strategists to do what the public says, it just forces them to get that outside opinion. Feedback in that sense is actually a good thing.

In fact, keeping up the actions might make negotiations with other countries easier, since there's a time element involved.

As for the killing of Ra, I'm not sure what the military rules are regarding the self-defense killing of a country ruler. I'm guessing that there are occasions when the life of a soldier is considered less important than the life of a foreign ruler. Maybe they had to do it but they could potentially be in deep trouble afterwards.