I'm not an expert but I feel that a more comfortable definition for me is to let $i: X\rightarrow G\times X$ by $x\mapsto (e,x)$ and define the action by $\Phi(a)s=i^*[(a\otimes 1)\cdot \phi^{-1}(\sigma^*s)]$. This illustrate the idea of "infinitesimal action of $a$".

@Adeel Yes, that's obviously my mistake. After reading your comment now I believe that maybe quasi-compact and quasi-separated is the best I can expect.

The Noetherian condition can be loosened to be locally Noetherian, isn't it?

Does my definition itself stand for other types of complexes?

@bananastack Yes. Actually for each $m$ we can take $\bigoplus_{0\leq i\leq -m}$ to be our strictly perfect complex in part one of the definition.

@FernandoMuro Yes smooth manifolds with the sheaf of smooth functions gives an affirmative example. However I'm looking for a more general criterion. For example, complex manifold in general does not satisfy my requirement. Only Stein manifolds work here. So I wonder what is the general condition here. I believe it's some cohomological requirement but I'm not sure.

@FernandoMuro Maybe that's not true, at least not so obvious, since for open set $U$, $V$, the complex of free sheaves $\mathcal{E}^{\bullet}_U$ and $\mathcal{E}^{\bullet}_V$ are quite different. Indeed they are quasi-isomorphic on $U\cap V$ but this does not guarantee that they glue together into a complex of locally free sheaves on $U\cup V$.

Thank you for pointing out! That is a typo and I have made correction in the definition of strictly perfect complex.