In the present paper, we revisit the geometry of smooth plane quartics and their bitangents from several perspectives. First, we study in detail the weak combinatorics of arrangements of bitangents associated with highly symmetric quartic curves. We consider quartic curves from the point of view of the order of their automorphism groups, in order to establish a lower bound on the number of quadruple intersection points for arrangements of bitangents associated with smooth plane quartics, which are smooth members of Ciani's pencil. We then construct new examples of $3$-syzygy reduced plane curves using smooth plane quartics and their bitangents.