Regarding (1) :
A) Every model category has an associated $\infty$-category, obtained for example by taking the Dwyer-Kan localization at the class of all equivalence, (But there are other more explicit way to construct it). It is always complete and co-complete, and is presentable when the model category is combinatorial.
Moreover Quillen functors induces adjoint functor between the associated $\infty$-category, and a Quillen functor is a Quillen equivalence if and only if it induces an equivalence of $\infty$-category.
B) Every presentable $\infty$-category is equivalent to the $\infty$-category associated to a combinatorial model category.
C) Given two combinatorial model categories, their associated $\infty$-category are equivalent if and only if the combinatorial category are connected by a zig-zag of Quillen equialence (In fact, a span of left Quillen functor is enough if I remember correctly).
The two results above are mostly deduced (with a bit of work) from Dugger's Theorem and the general theory of presentable $\infty$-category as it appears in Lurie's Higher topos theory.
D) The question of having a kind of model structure on the category of combinatorial model structure is open (or at least was until very recently, see F & F' below) and is listed as an open problem by Marc Hovey (both in his book on model categories and on his web page).
E) The question of what is the localization of the category of combinatorial model category at Quillen equivalences, and whether it is equivalent to the $\infty$-category of presentable $\infty$-category is also open. It has been discussed in the past on MO here and here, there are several interesting comment and answer on these questions that give partial result. There is also an nLab page on this problem.
F) Now, there has been some recent progress on (E) and (D), but this is mostly shameless self promotion, and even worse, about things I have not finish to writte yet, so what follow is barely an announcement:
I gave a talk at CT2019, where I claimed to construct three differents "right semi $2$-model structures" on the category of presentable categories endowed with two compatible combinatorial weak factorization systems, whose fibrant objects are respectively the "weak model structures", "the left semi-model structure" and the "left semi-model structure where every object is fibrant" and in each case the equivalences between fibrant objects are the Quillen equivalences.
Moreover, these three model structure are equivalent, and I can prove, using Karol Szumilo's result refered to in the question, that the $\infty$-categories attached to these model structure are all equivalent to the $\infty$-category of presentable $\infty$-categories.
The slides linked above contains more details about these model structures, how they are constructed, and the key idea in the proofs.
So this solve the problems mentioned in (D) & (E) above, at least when working with weak/left/right semi-model structures instead of Quillen model structures. I expect that it also really solves (E) for combinatorial Quillen model category as these form a nice subcategories of the model categories mentioned above, but I havn't really thought about this yet.
There also is a version of this story with simplicial model categories, and more generally enriched model categories, that is better behaved, and which actually gives Quillen $2$-model structures, instead of "right-semi".
F') I need to mention that Reid Barton has also developed very similar construction (completely independently, and possibly before mine) in his PhD thesis: His work do not cover everything I've mentioned before, and in particular do not say anything toward (E), but compare to mine has the big advantages of being already written (I do not know if it is already freely available though). His work construct the enriched version of what I call the "W" model structure in my slides, which is one of the three I've mentioned above, and show that it is actually a Quillen model structure (which is not true in the non-enriched case).
Regarding (2)
I think very little is known here. One thing one can do is to apply the version of Karol Szumilo's theorem depending on a cardinal $\kappa$, with $\kappa$ being the cardinal of a Grothendieck universes. This gives that (in the sense of this universe) co-complete large $\infty$-categories (non locally small) are equivalent to large cofibrations category with colimits of small chains of cofibrations.
I expect one can put back local smallness conditions by hand, to get an equivalence between locally small cocomplete categories and a special type of cofibration category, but I do not know of any result that cover the case of categories that are both complete & co-complete, not even of the kind of (B) and (C) above.
Also, as far as I know it is not known either what are the $\infty$-categories that can be represented by model categories in general.
