Serre's theorem on global generations on stacks

Tame Artin stacks (in the sense of Abramovich, Olsson and Vistoli, https://math.berkeley.edu/~molsson/tame.pdf) with quasi-projective moduli spaces will have property 2: the line bundle is the pullback of an ample line bundle on the moduli space.

As to property 1, a line bundle will not be enough, but you can get a version for quotient tame Artin stacks using a generating sheaf (in the sense of Olsson and Starr, https://math.berkeley.edu/~molsson/quot2a.pdf).

About 1 & 2, I doubt there are sensible results without some hypothesis like the existence an ample family of line bundles that makes the stack actually a scheme, in fact a so-called divisorial scheme.

As for 3, there is a paper that settles the issue, namely a quasi-compact and quasi-separated algebraic stack has affine stabilizer groups at closed points and satisfies the resolution property if and only if it is the quotient stack of a quasi-affine scheme by an action of $\mathop{GL}(n)$ for some $n$.

It is: Gross, Philipp: Tensor generators on schemes and stacks. Algebr. Geom. 4 (2017), no. 4, 501–522.

ArXiv version: https://arxiv.org/abs/1306.5418