I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint.

Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with integral coefficients. Furthermore, motives with transfers(!; see below) with rational coefficients "do not depend on the choice of the topology" (whether one takes Zar, Nis, Et, cdh, qfh, or h). Next, it is easier to prove "nice properties" of motives if one takes "the strongest possible" topology. Since one expects the etale realization of motives to be conservative, and etale cohomology satisfies h-descent, it is reasonable to consider h here.

As about $DM$ being isomorphic to $D^b(MM)$: one starts with the question of the existence of the motivic t-structure for DM. Its existence depends on the known relation of DM to the so-called motivic cohomology (and so, also to Chow motives) and on several hard conjectures; see the papers: Beilinson A., "Remarks on Grothendieck's standard conjectures"; Hanamura M., "Mixed motives and algebraic cycles, III".

To proceed further towards isomorphism to $D(MM)$ one needs the so-called $K(\pi,1)$-property for the motivic $t$-structure; I don't know any papers on this subject (yet you can look for them).

Now let's conside motives with integral coefficients. If one wants to the morphisms in DM to calculate the so-called motivic cohomology (= higher Chow groups) then et/qfh/h topologies should be abandoned. One can take Nis or even Zar instead; yet instead of "arbitrary" sheaves of abelian groups one should consider the so-called sheaves with transfers. Nis is a more convenient choice (and the cdh-topology is an important tool also).

I'm not sure that it is possible to compress the big picture into one answer; yet I will try to give a hint.

Firstly, one can hardly hope to have a "reasonable" motivic $t$-structure for motives with integral coefficients. Furthermore, motives with transfers(!; see below) with rational coefficients "do not depend on the choice of the topology" (whether one takes Zar, Nis, Et, cdh, qfh, or h). Next, it is easier to prove "nice properties" of motives if one takes "the strongest possible" topology. Since one expects the etale realization of motives to be conservative, and etale cohomology satisfies h-descent, it is reasonable to consider h here.

As about $DM$ being isomorphic to $D^b(MM)$: one starts with the question of the existence of the motivic t-structure for DM. Its existence depends on the known relation of DM to the so-called motivic cohomology (and so, also to Chow motives) and on several hard conjectures; see the papers: Beilinson A., "Remarks on Grothendieck's standard conjectures"; Hanamura M., "Mixed motives and algebraic cycles, III".

To proceed further towards isomorphism to $D(MM)$ one needs the so-called $K(\pi,1)$-property for the motivic $t$-structure; I don't know any papers on this subject (yet you can look for them).

Now let's consider motives with integral coefficients. If one wants the morphisms in DM to calculate the so-called motivic cohomology (= higher Chow groups) then et/qfh/h topologies should be abandoned. One can take Nis or even Zar instead; yet instead of "arbitrary" sheaves of abelian groups one should consider the so-called sheaves with transfers. Nis is a more convenient choice (and the cdh-topology is an important tool also).