Triangulated category associated to a subcategory of an abelian category

Question

As I know that if $\mathcal A_0$ is a thick subcategory of an abelian category $\mathcal A$, then I can define a triangulated subcategory $D^*_{\mathcal A_0}(\mathcal A )$ of $D^*(\mathcal A)$ ( $*= b, +,- )$ by taking all bounded complexes whose cohomology objects lie in $\mathcal A_0 $. And I have 2 questions:

1. Can we replace the condition on $\mathcal A_0$ of being 'thick' (closed under extensions) by another condition?

2. What if the category $\mathcal A$ is no more abelian ( quasi-abelian for example ? )

I will be so grateful too if anyone can give some reference to answer theses questions. Thank you in advance!