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@MikhailBondarko, it's not quite so easy, I'm afraid. You would need to be able to define Betti cohomology and Hodge structures at the level of dg-categories, something which is most likely impossible in my opinion. The best you have are noncommutative Hodge structures on Hochschild homology and variants, studied by Kaledin and others.
@MikhailBondarko, that's right, it was also proved directly in the note of Orlov (the proof essentially amounts to the same thing as the comparison of noncommutative motives modulo Tate twists with usual motives, namely Grothendieck-Riemann-Roch).
It's actually a conjecture of Orlov, see this paper. (The conjecture is for rational coefficients as it is clearly false integrally. I agree that it is not really expected to be true even rationally, though.)
@DmitryVaintrob, statements to this effect are in Blumberg-Gepner-Tabuada. They show that K-theory preserves filtered colimits and split exact sequences.
Any model category (in fact, even just a category with weak equivalences) gives rise to an $(\infty,1)$-category. For example, if one takes simplicially enriched categories as a model of $(\infty,1)$-categories, this is given by the Dwyer-Kan simplicial localization.
@ZhenLin: Gepner-Haugseng show that the $\infty$-category of $V$-enriched $\infty$-categories has an enrichment over itself, for $V$ a presentably $E_2$-monoidal $\infty$-category. In particular this gives the enrichment of DGCat over itself, and hence via Dold-Kan an enrichment over $\infty$-categories. (I'm using implicitly the rectification result of Haugseng, which implies that DGCat is equivalent to the $\infty$-category of $\infty$-categories enriched over the $\infty$-category of chain complexes.)
