In Bourbaki, Ch. VIII, $\S$ 7, Sect. 2, one can find the notion of an $\mbox{$R$-saturated set}$, and Corollary to Prop. 4 in that section proves that for every $R$-saturated set $\mathcal X$ there is a finite dimensional $\mathfrak g$-module whose set of weights coincides with $\mathcal X$. Prop. 6 in the next sections proves that the smallest $R$-saturated sets have the form $W.\lambda$ where $\lambda$ is minuscule. This answers Question (1). For Question (2) we take for $\mathcal X$ the union of $0$ and the set of all short roots in $R$. It is easy to see that this is an $R$-saturated set as verifying this reduces to root systems of rank 2 where everything is clear (even in type ${\rm G}_2$). Applying the above-mentioned Corollary we get a $\mathfrak g$-module with desired propoerties.

In Bourbaki, Ch. VIII, $\S$ 7, Sect. 2, one can find the notion of an $\mbox{$R$-saturated set}$, and Corollary to Prop. 4 in that section proves that for every $R$-saturated set $\mathcal X$ there is a finite dimensional $\mathfrak g$-module whose set of weights coincides with $\mathcal X$. Prop. 6 in the next sections proves that the smallest $R$-saturated sets have the form $W.\lambda$ where $\lambda$ is minuscule. This answers Question (1). For Question (2) we take for $\mathcal X$ the union of $0$ and the set of all short roots in $R$. It is easy to see that this is an $R$-saturated set as verifying this reduces to root systems of rank 2 where everything is clear (even in type ${\rm G}_2$). Applying the above-mentioned Corollary we get a $\mathfrak g$-module with desired properties. If $\beta$ is the dominant short root then, by construction, our set will coincide with the set of weigths of $L(\beta)$ as that set cannot be any smaller.