Yepper. Everyone knows if you put up a sign that says “No Guns Allowed”; everyone, including the nut case criminals, will obey the sign. They will double obey it if the state outlaws guns.
Not.
In the state of Washington if somebody has a sign like that in their store or business I just ignore it. And it is legal to do so. However, if they somehow discover that you have a firearm, they can ask you to leave. If you don’t, then you can be charged with trespassing.
Guns are banned from most schools, all bars, and the Post Office, courtrooms and other places like stadiums.