In this work, we employ fully atomistic molecular dynamics simulations to elucidate the effects of the oxidation pattern and of the water content on the organization of graphene sheets in aqueous dispersions and on the dynamic properties of the different moieties at neutral pH conditions. Analysis of the results reveals the role of the oxidation motif (peripherally or fully oxidized flakes) in the tendency of the flakes to self-assemble and in the control of key structural characteristics, such as the interlayer distance between the sheets and the average size and the distribution of the formed aggregates. In certain cases, the results are compared to a pertinent experimental system, validating further the relevant computational models. Examination of the diffusional motion of the oxidized flakes shows that different degrees of spatial restriction are imposed upon the decrease in the water content and elucidates the conditions under which a motional arrest of the flakes takes place. At constant water content, the structural differences between the formed aggregates appear to additionally impart distinct diffusional characteristics of a water molecule. A detailed examination of the counterion dynamics describes their interaction with the oxidized flakes and their dependence on the water content and on the oxidation pattern, offering new insight into the expected electrical properties of the dispersions. The detailed information provided by this work will be particularly useful in applications such as molecular sieving, nanofiltration, and in cases where conductive membranes based on oxidized forms of graphene are used.