That is true and the same is true for products. More than this, if your category has an enhancement that allows you to talk about homotopy co/limits, e.g., $T$ is the homotopy category of a stable model category, or of a bicomplete stable $(\infty,1)$-category or it is a base of a stable derivator, then not only the heart of any $t$-structure in $T$ is a bicomplete Abelian category, but any co/limit is computed as the $0$-th $t$-cohomology of the corresponding homotopy co/limit. For this, see Lemma 5.7 in this paper: https://arxiv.org/pdf/1708.07540.pdf

That is true and the same is true for products. More than this, if your category has an enhancement that allows you to talk about homotopy co/limits, e.g., $T$ is the homotopy category of a stable model category, or of a bicomplete stable $(\infty,1)$-category or it is a base of a stable derivator, then not only the heart of any $t$-structure in $T$ is a bicomplete Abelian category, but any co/limit is computed as the $0$-th $t$-cohomology of the corresponding homotopy co/limit. For this, see Lemma 7.3 in this paper: https://arxiv.org/pdf/1708.07540.pdf