Question 1 is not true in general. In characteristic 2 there are two symmetric representation-finite symmetric algebras with the same Auslander-Reiten quiver that are not isomorphic so one algebra has its Auslander algebra not given by mesh relations (and the relations are not even quadratic).

Question 2: There is now such an algorithm for standard representation-finite algebras, but not for non-standard. Not sure how to describe all non-standard ones and their Auslander algebra.