Hi,
(1. Seattle residents Zemke and Ray Hoffman met with city council members, who prompted the city to conduct a study on trash recycling.
2. The city chose a target of ...with the city council and mayor solidly behind the program.)
1. From the above two sentences, we can see that "the city" doesn't equal to "city council" , right? Then what does "the city" refer to? I don't think the word is intended by the writer to be a very precise expression. It's reallly just another way of saying 'the city council', although you could argue that the entire city, all its inhabitants, made the choice via their elected representatives.
2.What's the relationship between "the city council" and "mayor"? like the parliament and the president in the U.S.A.? No. For one thing, the US does not have a parliament. It has Congress, and the President is not 'the boss' of Congress, nor even a member of it. On the other hand, the Mayor is the 'boss', or perhaps more the leader, of the City Council. He is part of the Council. His actual powers can vary, from strong to weak.
Best wishes, Clive