0. This is a wide question. Probably the best answer/definition is precisely Bar Natan's one, which of course is implicit in Drinfeld's work: an associator is a filtered isomorphism between the completion of a naturally filtered, complicated, topological category, and a much more manageable, explicit, naturally graded, combinatorial one. In addition to the obvious interest in braid/knot theory, it turns out that the former is closely related to "quantum" objects, while the latter is related to "classical" objects. Hence associators shows up everywhere in deformation-quantization. Another "conceptual" explanation of this is that they are also responsible for the formality of the little disc operad, which is a close relative of the above isomorphism.

1. yes, if your category is $k[[\hbar]]$-linear for $k$ of char 0, this is basically the motivation behind the definition of GT. This is essentially the same things as the definition you gave, as the automorphism group of the completion of the braid category (as opposed to the automorphism group of the braid category, which is almost trivial).

2),3) yes, if the symmetric category is infinitesimal braided (i.e. if it's a representation of PaCD) then an associator turns it into a braided monoidal category (and even ribbon if it has duals).

4. Yes, it is exactly this: the Kontsevich integral for braids is exactly the above mentionned isomorphism, which extends to a functor from the category of tangles which is again an isomorphism after completion, and whose restriction to links is equal to the Kontsevich integral for any choice of associator.

All of this is explained in a nice, clear, pictorial way in many of Bar Natan's paper, especially of course those on Vassiliev invariants, and in Kassel--Turaev "Chord diagramms invariants of tangles and graphs".

0. This is a wide question. Probably the best answer/definition is precisely Bar Natan's one, which of course is implicit in Drinfeld's work: an associator is a filtered isomorphism between the completion of a naturally filtered, complicated, topological category, and a much more manageable, explicit, naturally graded, combinatorial one. In addition to the obvious interest in braid/knot theory, it turns out that the former is closely related to "quantum" objects, while the latter is related to "classical" objects. Hence associators shows up everywhere in deformation-quantization. Another "conceptual" explanation of this is that they are also responsible for the formality of the little disc operad, which is a close relative of the above isomorphism.

1. yes, if your category is $k[[\hbar]]$-linear for $k$ of char 0, this is basically the motivation behind the definition of GT. This is essentially the same things as the definition you gave, as the automorphism group of the completion of the braid category (as opposed to the automorphism group of the braid category, which is almost trivial).

2),3) yes, if the symmetric category is infinitesimal braided (i.e. if it's a representation of PaCD) then an associator turns it into a braided monoidal category (and even ribbon if it has duals).

4. Yes, it is exactly this: the Kontsevich integral for braids is exactly the above mentionned isomorphism, which extends to a functor from the category of tangles which is again an isomorphism after completion, and whose restriction to links is equal to the Kontsevich integral for any choice of associator.

edit: More precisely: as you say, associators are in bijection with isomorphisms $\widehat{PaB}\rightarrow PaCD$ compatible with the operadic structure. The Kontsevich integral provides such an isomorphism, hence produces a particular associator, namely the image of the trivial braid from $(\cdot \cdot) \cdot$ to $\cdot (\cdot \cdot)$ as explained in Bar-Natan's paper. This is the so-called KZ associator, which is the first (and for some time only) known associator. This is how Drinfeld showed that the set of associators in non-empty (and then deduced that there exists rational associators as well). Then this isomorphism extends to tangles, and by a result of Le-Murakami (implicit in Drinfeld's paper) its restriction to links doesn't depends on the choice of the associator (which proves the rationality of the Kontsevich integral of links).

All of this is explained in a nice, clear, pictorial way in many of Bar Natan's paper, especially of course those on Vassiliev invariants, and in Kassel--Turaev "Chord diagramms invariants of tangles and graphs".

0. This is a wide question. Probably the best answer/definition is precisely Bar Natan's one, which of course is implicit in Drinfeld's work: an associator is a filtered isomorphism between the completion of a naturally filtered, complicated, topological category, and a much more manageable, explicit, naturally graded, combinatorial one. In addition to the obvious interest in braid/knot theory, it turns out that the former is closely related to "quantum" objects, while the latter is related to "classical" objects. Hence associators shows up everywhere in deformation-quantization. Another "conceptual" explanation of this is that they are also responsible for the formality of the little disc operad, which is closely related to the above isomorphism.

1. yes, if your category is $k[[\hbar]]$-linear for $k$ of char 0, this is basically the motivation behind the definition of GT. This is essentially the same things as the definition you gave, as the automorphism group of the completion of the braid category (as opposed to the automorphism group of the braid category, which is almost trivial).

2),3) yes, if the symmetric category is infinitesimal braided (i.e. if it's a representation of PaCD) then an associator turns it into a braided monoidal category (and even ribbon if it has duals).

4. Yes, it is exactly this: the Kontsevich integral for braids is exactly the above mentionned isomorphism, which extends to a functor from the category of tangles which is again an isomorphism after completion, and whose restriction to links is equal to the Kontsevich integral for any choice of associator.

All of this is explained in a nice, clear, pictorial way in many of Bar Natan's paper, especially of course those on Vassiliev invariants, and in Kassel--Turaev "Chord diagramms invariants of tangles and graphs".

0. This is a wide question. Probably the best answer/definition is precisely Bar Natan's one, which of course is implicit in Drinfeld's work: an associator is a filtered isomorphism between the completion of a naturally filtered, complicated, topological category, and a much more manageable, explicit, naturally graded, combinatorial one. In addition to the obvious interest in braid/knot theory, it turns out that the former is closely related to "quantum" objects, while the latter is related to "classical" objects. Hence associators shows up everywhere in deformation-quantization. Another "conceptual" explanation of this is that they are also responsible for the formality of the little disc operad, which is a close relative of the above isomorphism.

1. yes, if your category is $k[[\hbar]]$-linear for $k$ of char 0, this is basically the motivation behind the definition of GT. This is essentially the same things as the definition you gave, as the automorphism group of the completion of the braid category (as opposed to the automorphism group of the braid category, which is almost trivial).

2),3) yes, if the symmetric category is infinitesimal braided (i.e. if it's a representation of PaCD) then an associator turns it into a braided monoidal category (and even ribbon if it has duals).

4. Yes, it is exactly this: the Kontsevich integral for braids is exactly the above mentionned isomorphism, which extends to a functor from the category of tangles which is again an isomorphism after completion, and whose restriction to links is equal to the Kontsevich integral for any choice of associator.

All of this is explained in a nice, clear, pictorial way in many of Bar Natan's paper, especially of course those on Vassiliev invariants, and in Kassel--Turaev "Chord diagramms invariants of tangles and graphs".