As a physicist, I have some naive questions about mixed motives and its mixed Hodge structure (MHS) realization. Any references, comments, answers will be appreciated!

The category of mixed motives over $\mathbb{Q}$ has not been constructed, but anyway let us suppose it exists and denote it by $\mathcal{MM}_{\mathbb{Q}}$, and I want to know some expected properties of it. Every mixed motive $M$ then has a Hodge realisation, denoted by $H(M)$, which is a MHS (an obeject in the abelian category of \mathbb{Q}-MHS), i.e. a functor \begin{equation} H:\mathcal{MM}_{\mathbb{Q}} \rightarrow \mathbb{Q}\,MHS \end{equation} First, suppose $H(M)$ is the direct sum of $S_1$ and $S_2$ in the category $\mathbb{Q}$-MHS, do we expect there exist $M_1$ and $M_2$ in $\mathcal{MM}_{\mathbb{Q}}$ such that $H(M_i)=S_i$ and $M=M_1 \oplus M_2$? If yes, does this property have anything to do with Hodge conjecture?

Second, suppose there is a sequence in $\mathcal{MM}_{\mathbb{Q}}$, \begin{equation} 0\rightarrow M_1 \rightarrow M \rightarrow M_2 \rightarrow 0 \end{equation} which we do not require to be exact. If its Hodge realization is exact in the category $\mathbb{Q}$-MHS, i.e. the following sequence is exact, \begin{equation} 0\rightarrow H(M_1) \rightarrow H(M) \rightarrow H(M_2) \rightarrow 0 \end{equation} Do we expect the sequence upstairs is exact!

Third, Veovodsky has constructed a triangulated category which is candidate for the derived category of the assumed category $\mathcal{MM}_{\mathbb{Q}}$, denote it by $\mathcal{DMM}_{\mathbb{Q}}$, does there exist a functor which looks like a Hodge realization functor? i.e. a functor \begin{equation} \widetilde{H}:\mathcal{DMM}_{\mathbb{Q}} \rightarrow \mathbb{Q}\,MHS \end{equation}

Fourth, if third is true, $\widetilde{H}$ exists, is a similar property like (First) true when we replace $\mathcal{MM}_{\mathbb{Q}}$ by $\mathbb{DMM}$? Similarly, is there a similar property like (second) when we replace $\mathcal{MM}_{\mathbb{Q}}$ by $\mathbb{DMM}$? (need to replace SES by a distinguished triangle in the question)?