1. Yes.
2. Not sure what it's called, but there is an advisor (foreign advisor?) that shows if civs are at war, have peace, have met, have open borders, are vassals (Warlords and BTS), or have defense pact/permanent alliance. This can give you a clue as to who cares about who.
3. Usually coastal cities will be commerce oriented. The Moai Statues National Wonder will give all sea tiles IN THAT CITY ONLY hammers. (National meaning it can be built by each nation, but only in one city each. However, no more that 2 national wonders can be built in each city.)
4. They are unreliable recommendations by the computer. When your worker is in them, some of the icons will blink blue. These are the computer's reccomendations of what you should build, for instance on a rice tile the farm icon will blink.
5. The shrine can only be destroyed if you destroy the holy city, and since shrines can only be built in the holy city, they cannot be rebuilt. So as long as you leave the holy city with a shrine in it intact, it is yours.
Shrines give +1 commerce

commerce

for every city that practices that religion.
The only way to wipe out a religion is to burn all the cities that practice that religion.