@TimCampion It's not a deep fact. You can find a discussion in paragraph 2.2 here.

@JacobGross: It is true that given a derived moduli problem, you can ask whether it admits a cotangent complex and whether it is of tor-amplitude $[-1,0]$, in which case it would admit a virtual fundamental class. However you need first of all to extend your classical moduli problem to a derived moduli problem in a reasonable way, which may not always be trivial at all.

One can always define the category of quasi-coherent sheaves on a (derived) stack $X$ as the limit $\varprojlim_{S\to X} Qcoh(S)$ over all affine schemes $S$ over $X$. If $X$ is an Artin stack (resp. $n$-Artin derived stack) then this can be described using the lisse-étale site instead, i.e. take the limit only over smooth morphisms $S \to X$; see e.g. Prop. 1.4.2 here. This is equivalent to the description of Qcoh given in Olsson's book essentially for formal reasons.

You might want to look at § 7.2 of Lurie's Higher Algebra. For example, coherent = almost perfect + eventually coconnective ($\pi_i=0$ for $i \gg 0$). Your suggestion about finite tor-dimension is incorrect: the condition that $M \otimes_{O_X} -$ commutes with products is the same (assuming you mean derived tensor product) as dualizability (which is the same as perfectness).