For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it:
(a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here one can either take the "global" or "Nisnevich-local" homotopy category;
(b) the homotopy category of simplicial Nisnevich sheaves;
(c) the unstable motivic homotopy category;
(d) the motivic homotopy category of $S^1$-spectra;
(e) the motivic homotopy category of $T$-spectra;
(f) the category of highly structured modules over the Voevodsky spectrum $MGl$;
(g) the ("big") category of Voevodsky motives over $k$.
I would like to have some references about natural connecting functors between these categories and their models. In particular, I am interested in the following questions.
1) Do the natural connecting functors from (c) into (d) and from (d) into (e) (and possibly, further in the list in any direction) lift to left Quillen functor between certain models of these categories (I would prefer to start from the injective model structure for (c)).
2) Does there exist an exact functor from (e) into (g) that sends the $T$-spectrum of a smooth $k$-variety into its motif? If $k$ is of characteristic $0$ then the main result of "Modules over motivic cohomology" certainly gives a functor of this sort. I was not able to understand whether this paper gives an answer to my question in general.