Dear Andreas Martin Lisewski,
many thanks for the very detailed and interesting questions. -- Let me begin with the second one:
The whole point of "emergent" quantum mechanics is to show that there is not necessarily such conceptual leap between quantum mechanics and "classical" theories; but that quantum mechanics appears to us in an analogous way, through coarse graining, as hydrodynamics emerges from atomic physics. This point of view has recently found much attention by, for example, L Smolin and F Markopoulou, S Adler, G 't Hooft, G Vitiello with M Blasone and P Jizba, myself, and others. Papers can be found online in the "arXiv". In the present article I show, as a result, and with reference to more detailed work, that the Liouville equation and the von Neumann equation do not need a different conceptual starting point, but that they are related to each other by a unique set of tranformations -- AND -- differ from each other, in a suitable representation, by one characteristic term.
The aim of the rather condensed last section in the present paper is to discuss that the causal / extensional set perspective provides heuristic arguments which allow to understand under which conditions the Liouville equation does go over to the von Neumann equation, by eliminating the term by which they could differ. This is NOT to say that this is all of QM! I indicated this, open questions in particular concerning Born rule, the danger or interesting aspect of negative probabilities etc., some of which are discussed in more detail in the references given. What is obtained here is the dynamical equation of QM, in the usual Hilbert space language.
To the first question: Once interactions are introduced, in order to continue the present first step by providing a dynamical "toy model", one certainly should be able to see under which conditions the duality is sort of maximally broken, which would provide a causal set theory of a spacetime -- as much studied by R Sorkin and collaborators, see references in the article. One of the aims of the causal set research program, then, is to understand how large dynamical causal sets may approach a continuum limit, and particularly one which is described by GR.
Thus, concerning both your questions, QM and GR possibly arise as limiting cases of a more encompassing theory, in which causality and extensity, besides discreteness, are believed to be essential ingredients.
I hope, these remarks help to place my article in perspective of currently active research topics.
Thank you again and my best regards,
Hans-Thomas Elze