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). 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 cocomplete and finitely complete $\infty$-category is complete and cocomplete but neither presentable nor copresentable. In "Higher Topos Theory" Lurie considers the pro category of spaces (see Definition 7.1.6.1). It thus seems that this category gives a positive answer to (1) 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.

**Edit:** It turns out that there is a mistake in [Raptis's paper][1] showing that under Vopenka's principle any cofibrantly generated model category is Quillen equivalent to a combinatorial one. Raptis and Rosicky posted a [fixed proof][2], in which they had to assume that the domains of the generating (acyclic) cofibrations (in the cofibrantly generated model category) are small with respect to certain types of filtered colimits of cofibrations, more general than just chains of cofibrations. See also the remarks of Tim Campion below.  

[1]: https://arxiv.org/abs/0907.2726
[2]: https://arxiv.org/abs/1703.08094