1. Yes, they do. Most of it has to do with the ratings impacts of how you treat (promote, demote, etc.) them, as well as filing for FA after going to arbitration too many times.
2. Yes, but a good eye seems to affect other things indirectly.
3. A far bit, but what you describe is above average, imho.
PS: You can often get a quicker response with a new thread, or replying to one of the "live" ones below. This is mostly here for a reference.