Let $G$ be a reductive group. If one can associate to $G$ a shimura datum $(G,X)$, then the étale cohomology of the associated Shimura variety $\operatorname{Sh}(G,X)$ is a strong tool for Langlands correspondence because it carries an action of $G(\mathbb{A})\times\operatorname{Gal}(\mathbb{Q}^{\operatorname{al}},\mathbb{Q})$.

(It is of course more complicated, one should first show that these Shimura variety admit a model over a number field but this known in general using the theory of special points)

I have recently heard that some group can't admit a Shimura datum, so my questions are

0- Is this really true?

1- Is there a criterion to know if $G$ can admit a Shimura datum?

2- Is there a criterion to know if $G$ can't admit a Shimura datum?

3- If $G$ don't have any Shimura datum, then with what to replace Shimura variety? Can Flag varieties be helpful for example?