We use morphological measurements and the scatter of clusters about observed and simulated scaling relations to examine the impact of merging and core-related phenomena on the structure of galaxy clusters. All relations constructed from emission-weighted mean temperature and intracluster medium mass, X-ray luminosity, isophotal size, or near-IR luminosity show a separation between cool core (CC) and non-cool core (NCC) clusters. We attribute this partially to a temperature bias in CC clusters, and partially to other cool core-related structural changes. We attempt to minimize CC/NCC separation in scaling relations by applying a uniform scale factor to CC cluster temperatures and determining the scale factor for each relation that minimizes the separation between CC and NCC populations, and by introducing central surface brightness as a third parameter in relations. The latter approach reduces scatter in relations more than temperature scaling. We compare the scatter within subsamples split by CC/NCC and morphological merger indicators. CC clusters and clusters with less substructure generally exhibit higher scatter about relations. The larger structural variations in CC clusters exit well outside the core, suggesting that a process more global than core radiative instability is at work. Simulations without cooling mechanisms also show no correlation between substructure and larger scatter about relations, indicating that any merger-related scatter increases are subtle. The results indicate that cool core related phenomena, not merging processes, are the primary contributor to scatter in scaling relations. Our analysis does not appear to support the scenario in which clusters evolve cool cores over time unless they experience major mergers. (Abridged)