Ill take a shot at it (no pun intended):
1. Generally, no. Reference:
RSMO 571.030. 1. A person commits the crime of unlawful use of weapons if he or she knowingly: (8) Carries a firearm or any other weapon readily capable of lethal use into any church or place where people have assembled for worship, or into any election precinct on any election day, or into any building owned or occupied by any agency of the federal government, state government, or political subdivision thereof;
If you are going to carry into a city library, that falls into this category. I believe if a the state or the federal government contributes funding for any library then it will fall into this category, though I have no reference for that. Maybe someone else can correct me or solidify that.
2. Technically, yes. If the sign is the only indication (for example, there is nothing published and theres no other verbiage elsewhere) and it says "No Concealed Weapons" then your good. Its up to you, some people may see this as a gray area, and think you are pressing your luck, based on "the spirit" of the sign, but by definition open carry is not concealed.
3. Is a frog's poo hole watertight? Conditionally, yes. Based on the unfortunate inability to predict what is going to happen in the near future it is crucial to be as prepared as humanly possible. However, due to limitations in training and/or equipment you may decide not to. Carrying chambered is a personal choice, and if you feel like you are more likely to have a ND on a draw because of a chambered round, then maybe its not for you. If you have a holster that has little or no retention then you may decide this is not for you. If you think your weapon may go off if dropped, don't drop it and buy a better gun. I carry my Glock chambered in a Glaco thumb break and I have no concerns about safety. I carry a 1911 in a Serpa and have no concern about safety. When I carried my Glock in a Fobus (dumb idea, I know) I had a round chambered about 30% of the time. With no retention, I wanted that couple seconds of "oh sh**" time while the bad guy fumbled with my gun to try and get it back or run away. Then I decided I was stupid and got a better holster. Now my Fobus is a chew toy for my teething son. Whatever you are most comfortable and competent with.