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.
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).
- 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".