The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $ y $ satisfying some pentagon and hexagon relations.

In other words, denoting by $ \mathfrak{lie}_2 $ the free Lie algebra on two variables, an associator is a grouplike element in the completed hopf algebra $ \widehat{U}(\mathfrak{lie_2})$ satisfying the above mentionned relations.

In more abstract terms, there is a bijection between the set of associators and a certain morphism of operads. Using Bar-Natan's notations it is a morphism of operads between $ \widehat{PaB_K} $ and $ PaCD $.

In this context, the Grothendieck-Teichmüller group $ \widehat{GT} $ is the group of automorphism of $ \widehat{PaB_K} $. It can also be described as pairs $ (f,\lambda) \in \widehat{F_2(K)} \times K^* $ again satisfying some relations.

These seem to be the "textbook" definitions, however I don't understand their motivation and what these objects do. My questions are the following. (The lack of precision comes from my lack of knowledge of the subject)

0) What is the purpose of associators? What is the purpose of the G-T group in this context (not in algebraic geometry).

The following questions are possible answers to the previous one.

1) Let $ (C, \beta, \gamma) $ be a braided monoidal category with associativity constraint $ \gamma $ and braiding $ \beta $. Suppose we wish to change $ \beta $ and $ \gamma $ into $ \beta' $ and $ \gamma' $ such that $ (C, \beta', \gamma') $ is again a braided monoidal category. Does this procedure have a name? Do elements of the Grothendieck-Teichmüller control this?

2) Let $ (C, \sigma, \gamma) $ be a symmetric monoidal category. Do the associators control how to transform $ C $ into a braided monoidal category?

3) Does the term infinitesimally braided monoidal category exist?

4) Does the Kontsevich integral somehow fit into this story?

As I mentioned above, I know little about the subject: I have skimmed through Bar-Natan's article "On associators and the Grothendieck-Teichmüller group group" and the paper arXiv:0903.4067, therefore any reference is welcome.