Here is a simple way to talk about the homotopy type of a stack. Let $\mathfrak{X}$ be a stack and $f: U \to \mathfrak{X}$ a representable surjective submersion (a covering) from a space $U$ (e.g., the map from the Teichmuller space to the moduli stack.) Now, form the pullback of $f$ along itself: $U\times_{\mathfrak{X}} U$. This comes with two maps to $U$ and there is a diagonal map from $U \to U\times_{\mathfrak{X}} U$. All together, these maps give a topological groupoid. The nerve of this topological groupoid is a simplicial space and the geometric realization of which is a space that one can regard as representing the homotopy type of the stack.

Here are some easy/nice properties of the above notion of homotopy type that are easy to check.

1. Given a space $X$ with a $G$ action, the homotopy type of $[X/G]$ is the Borel construciton, aka homotopy quotient, $EG \times_G X$. In particular, the answer to your Question 1 is affirmative, and the homotopy type of the moduli stack of curves is exactly $B\Gamma_g$.

2. One can define singular and de Rham cohomology of a stack and these invariants coincide with the integral and rational cohomology of the homotopy type of the stack. This is in fact almost a tautology since, for example, the de Rham cohomology can be defined by taking a covering by a manifold, forming iterated pullbacks (to produce a simplicial manifold), taking the de Rham algebra of this simplicial manifold to get a cosimplicial dga, and then taking the totalization to get a dga.

3. It follows from property 1 above that the answer to your question 2 is also affirmative.

If I remember correctly, Noohi uses a slightly more sophisticated notion of homotopy type. He defines a universal weak equivalence to be a representable morphism from a space $U$ to a stack $\mathfrak{X}$ such that the pullback along any morphism from a space to $\mathfrak{X}$ is a weak equivalence. $U$ can then be regarded as the homotopy type of the stack. I think this is more of less equivalent to the naive version I explained above, but it has the advantage of being a bit more functorial and there might be some other technical advantages I can't remember. David Carchedi will probably be able to give more details.

Here is a simple way to talk about the homotopy type of a stack. Let $\mathfrak{X}$ be a stack and $f: U \to \mathfrak{X}$ a representable surjective submersion (an atlas) from a space $U$ (e.g., the map from the Teichmuller space to the moduli stack.) Now, form the pullback of $f$ along itself: $U\times_{\mathfrak{X}} U$. This comes with two maps to $U$ and there is a diagonal map from $U \to U\times_{\mathfrak{X}} U$. All together, these maps give a topological groupoid. The nerve of this topological groupoid is a simplicial space and the geometric realization of which is a space that one can regard as representing the homotopy type of the stack.

Here are some easy/nice properties of the above notion of homotopy type that are easy to check.

1. Given a space $X$ with a $G$ action, the homotopy type of $[X/G]$ is the Borel construciton, aka homotopy quotient, $EG \times_G X$. In particular, the answer to your Question 1 is affirmative, and the homotopy type of the moduli stack of curves is exactly $B\Gamma_g$.

2. One can define singular and de Rham cohomology of a stack and these invariants coincide with the integral and rational cohomology of the homotopy type of the stack. This is in fact almost a tautology since, for example, the de Rham cohomology can be defined by taking a covering by a manifold, forming iterated pullbacks (to produce a simplicial manifold), taking the de Rham algebra of this simplicial manifold to get a cosimplicial dga, and then taking the totalization to get a dga.

3. It follows from property 1 above that the answer to your question 2 is also affirmative.

4. [Edit] This notion of homotopy type is well-defined because one can check that any two atlases determine Morita equivalent topological groupoids which then have weakly equivalent nerves.

If I remember correctly, Noohi uses a slightly more sophisticated notion of homotopy type. He defines a universal weak equivalence to be a representable morphism from a space $U$ to a stack $\mathfrak{X}$ such that the pullback along any morphism from a space to $\mathfrak{X}$ is a weak equivalence. $U$ can then be regarded as the homotopy type of the stack. I think this is more of less equivalent to the naive version I explained above, but it has the advantage of being a bit more functorial and there might be some other technical advantages I can't remember. David Carchedi will probably be able to give more details.