Use of Tiles around cities

I have a couple questions about how the binding of tiles around cities are used and manipulated.

It appears that some of the tiles around my cities, that were already in use by my city, have been stolen from my city by another player.  Is this really possible, and how is it done?  And how can I get them back?

And what is the second funtion listed on the Workers toolbox that is a picture of a sphere with a ~ in the middle of it?

Also, how can the quality of tiles that we bind to our cities be improved from A to U to C to H, thereby bringing them into full use?  I have attempted to improve their quality by increasing Luxury; adding them in by their levels of Productivity, Food, and/or Travel; and even by changing type of Government.  I can't determine the patern.  What are the contributing factors that we can use to improve?