For the geometric aspects in the first question, Alan Durfee's survey linked by Sandor is a good starting point. For the second question, it's probably necessary to get into the viewpoint of algebraic groups (in characteristic 0). Grothendieck foretold where the simple singularities should occur "in nature" in the algebraic group framework. This was proved by Brieskorn, whose student Peter Slodowy then filled in the picture. Slodowy's monograph *Simple Singularities and Simple Algebraic Groups* (Springer Lect. Notes. in Math. 815, 1980) is still a basic source, along with his various lecture notes and later papers. Here one sees the Dynkin diagrams intrinsically realized. At the same time the whole subject is treated in suitable generality in terms of the group structure.

Maybe it's possible to understand your first question without going this far into the technical aspects of algebraic groups, but it's good to be aware of that direction along with traditional approaches to singularities.