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If you believe Vopěnka's principle then any cofibrantly generated model category is Quillen equivalent to a combinatorial one and thus its underlying $\infty$-category is presentable. It follows that any non-presentable complete and cocomplete $\infty$-category would give a positive answer to (1) and (2). The pro-category of a small finitely complete $\infty$-category is not presentable but it is co-presentable (its opposite category is presentable). However, the pro category of a large finitely complete and cocomplete $\infty$-category is complete and cocomplete but neither presentable nor copresentable. In "Higher Topos Theory" Lurie considers the pro category of spaces (when defining the shape of topoi). Is thus seems that this category gives a positive answer to (1), (2) and (3). This example was also considered in the world of model categories: It is Isaksen's strict model structure on pro-simplicial sets. If you are interested in a cocombinatorial model category that arose naturally you have Morel's (resp. Quick's) model structure on simplicial pro-finite sets that models the $\infty$-category of p-pro-finite (resp. pro-finite) spaces.