Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following  questions.

 

 1. When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some Lie group $G$?
 2.  When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(M\rightrightarrows M)$ for some manifold $M$?
 3.  When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $G\ltimes M$ for some Lie group $G$ acting on a manifold $M$?

It is only reasonable to see what properties the Lie groupoids $(G\rightrightarrows *), (M\rightrightarrows ), G\ltimes M$ have so that we can say $\mathcal{G}$ should have atleast these properties.

My observations: 

 1. Given a Lie group $G$, the Lie groupoid $(G\rightrightarrows *)$ is transitive Lie groupoid. Thus, it should be the case that $\mathcal{G}$ is necessarily transitive to be represented by Lie group. Surprisingly, any transitive Lie groupoid is representable by a Lie group, more precisely $\mathcal{G}\cong \mathcal{G}_x$ (isotropy group) for any $x\in \mathcal{G}_0$ (whose proof I do not know). I do not know if we can say anything more in this. Is there any other characterization? Can we say anything more?
 2. For a manifold $M$, the Lie groupoid $(M\rightrightarrows M)$ is a proper Lie groupoid simply because, $(s,t):M\rightarrow M\times M$ is the diagonal map which is a proper map. So, $(M\rightrightarrows M)$ is a proper Lie groupoid. If we want $\mathcal{G}$ to be represented by manifold, we should have atleast this condition that $\mathcal{G}=(P\rightrightarrows X)$ is a proper Lie groupoid i.e., $(s,t):P\rightarrow X\times X$ is a proper map. For manifold $M$, $(s,t):M\rightarrow M\times M $ is injective. If we want $\mathcal{G}$ to be represented by manifold, we should have atleast this condition that $\mathcal{G}=(P\rightrightarrows X)$ is such that $(s,t):P\rightarrow X\times X$ is injective. David Roberts says [here][1] that, assuming  that $(s,t):P\rightarrow X\times X$ is proper and injective sufficient to  confirm $\mathcal{G}\cong X/P$ (Any quick proof for this is also welcome). Are there any other characterization?   How do we know where to stop looking at properties for characterization? I mean how do we guess (before proving) that $(s,t)$ is proper and injective then, $\mathcal{G}$ is represented by manifold. Why **proper, injective** determining "manifoldness" of Lie groupoid?


Are there similar characterizations for a Lie groupoid to be a translation groupoid i.e., of the form $G\ltimes M$ for some Lie group $G$ acting on manifold $M$? It is observed [here][2] that, if a Lie groupoid is proper and etale, then $\mathcal{G}$ is *locally* a translation groupoid. Are there any characterization for whole Lie groupoid to be translation groupoid? What should we look for in a Lie groupoid to expect it to be a translation groupoid?
 


  [1]: https://mathoverflow.net/questions/308306/to-check-if-a-stack-is-coming-from-a-manifold
  [2]: https://mathoverflow.net/questions/303183/proper-and-etale-groupoid-is-locally-a-translation-groupoid