We study the fraction of tidal interactions and mergers (TIMs) with well-identified observability timescales (f(TIM)) in group, cluster, and accompanying field galaxies and its dependence on redshift (z), cluster velocity dispersion (sigma), and environment analyzing Hubble Space Telescope/ACS images and catalogs from the ESO Distant Cluster Survey. Our sample consists of 11 clusters, seven groups, and accompanying field galaxies at 0.4 <= z <= 0.8. We derive f(TIM) using both a visual classification of galaxy morphologies and an automated method, the G - M-20 method. We calibrate this method using the visual classifications that were performed on a subset of our sample. We find marginal evidence for a trend between f(TIM) and z, in that higher z values correspond to higher f(TIM). However, we also cannot rule out the null hypothesis of no correlation at higher than 68% confidence. No trend is present between f(TIM )and sigma. We find that f(TIM) shows suggestive peaks in groups, and tentatively in clusters at R > 0.5 x R-200, implying that f(TIM) gets boosted in these intermediate-density environments. However, our analysis of the local densities of our cluster sample does not reveal a trend between f(TIM) and density, except for a potential enhancement at the very highest densities. We also perform an analysis of projected radius-velocity phase space for our cluster members. Our results reveal that TIM and undisturbed galaxies only have a 6% probability of having been drawn from the same parent population in their velocity distribution and 37% in radii, in agreement with the modest differences obtained in f(TIM) at the clusters.