if for example i set my city to focus on production
does it means that the automated workers will focus on building +:hammers: around it?
overall what does effect the AI automated workers to build, lets say a farm instead of a cottage?

My understanding is that the city manager is independent of the automated worker.

If you tell the city manager to focus on production, it will shuffle the citizens to work the production tiles based on how they are currently improved.

The automated worked improves the tiles based on what the game sees as the best improvement for that terrain.

Automated workers will in fact listen to the city governor's suggestions, so it is exactly as you say: if you have the city governor on :hammers: then workers will build +:hammers:. This is NOT true in vanilla Civ4, but was one of the many AI improvements added in Beyond the Sword.