Sorry for answering so late, I just was pointed to this question. Before answering your questions in the case of a Lie group (i.e., one object) let me point out that the difference between $\mathbb{R}$-coefficients and $\mathbb{S}^1$-coefficients it not negligible but is somehow the whole heart of the story.

1. The cohomology of the continuous cochains is equal to the cohomology of the smooth cochains. Moreover, there is yet another complex $C^n_{ec}$ of cochains that are continuous around $\Delta^{(n)}$. Then $H^n_{ec}$ equals $H^n_{es}$ (all this for coefficients in $\mathbb{R}$, $\mathbb{S}^1$ or more general tori).

2. The universal cover $\mathbb{Z}\to\mathbb{R}\to\mathbb{S}^1$ is described by a cocycle in $Z^2_{es}$ that cannot be smooth (otherwise $ \mathbb{R}\to\mathbb{S}^1$ would have a smooth global section) See 3.

3. $H^n_{es}=H^n$ (in your notation form above) if either the Lie group $G$ or the coefficients are contractible (in particular the difference between $\mathbb{R}$ and $\mathbb{S}^1$ plays a significant role).

A reference for the equality of smooth and continuous cohomology is Hochschild, Mostow, "Cohomology of Lie groups" and for the more general case this paper, together with F. Wagemann.

I expect that the case of general Lie groupoids can be treated similarly, for instance I would expect that $H^n=H^n_{es}$ if the Lie groupoid $\Gamma$ is topologically trivial (i.e., its simplicial manifold $N\Gamma$ is contractible). However, the generalisation of our approach is not entirely straight forward.

Sorry for answering so late, I just was pointed to this question. Before answering your questions in the case of a Lie group (i.e., one object) let me point out that the difference between $\mathbb{R}$-coefficients and $\mathbb{S}^1$-coefficients it not negligible but is somehow the whole heart of the story.

1. The cohomology of the continuous cochains is equal to the cohomology of the smooth cochains. Moreover, there is yet another complex $C^n_{ec}$ of cochains that are continuous around $\Delta^{(n)}$. Then $H^n_{ec}$ equals $H^n_{es}$ (all this for coefficients in $\mathbb{R}$, $\mathbb{S}^1$ or more general tori).

2. The universal cover $\mathbb{Z}\to\mathbb{R}\to\mathbb{S}^1$ is described by a cocycle in $Z^2_{es}$ that cannot be smooth (otherwise $ \mathbb{R}\to\mathbb{S}^1$ would have a smooth global section), see also 3.

3. $H^n_{es}=H^n$ (in your notation form above) if either the Lie group $G$ or the coefficients are contractible (in particular the difference between $\mathbb{R}$ and $\mathbb{S}^1$ plays a significant role).

A reference for the equality of smooth and continuous cohomology is Hochschild, Mostow, "Cohomology of Lie groups" and for the more general case this paper, together with F. Wagemann.

I expect that the case of general Lie groupoids can be treated similarly, for instance I would expect that $H^n=H^n_{es}$ if the Lie groupoid $\Gamma$ is topologically trivial (i.e., its simplicial manifold $N\Gamma$ is contractible). However, the generalisation of our approach is not entirely straight forward.