The answer to question 1 is “yes” since every weak equivalence is essentially surjective (let $y\in Y$, $[y]$ the connected component of $y$, which is in the image of $\pi_0(F)$, thus there exists $y^\prime\in [y]$ which is an element of the image of $F$ and isomorphic to $y$) and full and faithful (since $F_x$ is an isomorphism, the mappings $F_{xy}\colon A(x,y)\to B(F(x),F(y))$ are also bijective, choose $f\in A(y,x)$ then $\alpha\colon A(x,y)\to A(x), g\mapsto fg$ and $\beta\colon B(F(x),F(y))\to B(F(x)), g\mapsto F(f)g$ are bijections and $F_{xy}=\beta^{-1}\circ F_x\circ \alpha$ is bijective).

As a counter example for question two consider: $\Gamma=C_2$, $X$ the “complete” (in the sense of graph theory) groupoid consisting of two objects and four isomorphisms. The action of $\Gamma$ on $X$ is defined to swap the objects. Let $Y$ be the terminal groupoid consisting of a single isomorphism. $\Gamma$ acts trivially on $Y$. The unique morphism $X\to Y$ is a weak equivalence of $\Gamma$-groupoids. But there does not exist a $\Gamma$-morphism $Y\to X$.

The answer to question 1 is “yes” since every weak equivalence is essentially surjective (let $y\in Y$, $[y]$ the connected component of $y$, which is in the image of $\pi_0(F)$, thus there exists $y^\prime\in [y]$ which is an element of the image of $F$ and isomorphic to $y$) and full and faithful (since $F_x$ is an isomorphism, the mappings $F_{xy}\colon A(x,y)\to B(F(x),F(y))$ are also bijective, choose $f\in A(y,x)$ then $\alpha\colon A(x,y)\to A(x), g\mapsto fg$ and $\beta\colon B(F(x),F(y))\to B(F(x)), g\mapsto F(f)g$ are bijections and $F_{xy}=\beta^{-1}\circ F_x\circ \alpha$ is bijective).

As a counter example for question 2 consider: $\Gamma=C_2$, $X$ the “complete” (in the sense of graph theory) groupoid consisting of two objects and four isomorphisms. The action of $\Gamma$ on $X$ is defined to swap the objects. Let $Y$ be the terminal groupoid consisting of a single isomorphism. $\Gamma$ acts trivially on $Y$. The unique morphism $X\to Y$ is a weak equivalence of $\Gamma$-groupoids. But there does not exist a $\Gamma$-morphism $Y\to X$.

The answer to question 1 is “yes” since every weak equivalence is essentially surjective (let $y\in Y$, $[y]$ the connected component of $y$, which is in the image of $\pi_0(F)$, thus there exists $y^\prime\in [y]$ which is an element of the image of $F$ and isomorphic to $y$) and full and faithful (since $F_x$ is an isomorphism, the mappings $F_{xy}\colon A(x,y)\to B(F(x),F(y))$ are also bijective, choose $f\in A(y,x)$ then $\alpha\colon A(x,y)\to A(x), g\mapsto fg$ and $\beta\colon B(F(x),F(y))\to B(F(x)), g\mapsto F(f)g$ are bijections and $F_{xy}=\beta^{-1}\circ F_x\circ \alpha$ is bijective).

The answer to question 1 is “yes” since every weak equivalence is essentially surjective (let $y\in Y$, $[y]$ the connected component of $y$, which is in the image of $\pi_0(F)$, thus there exists $y^\prime\in [y]$ which is an element of the image of $F$ and isomorphic to $y$) and full and faithful (since $F_x$ is an isomorphism, the mappings $F_{xy}\colon A(x,y)\to B(F(x),F(y))$ are also bijective, choose $f\in A(y,x)$ then $\alpha\colon A(x,y)\to A(x), g\mapsto fg$ and $\beta\colon B(F(x),F(y))\to B(F(x)), g\mapsto F(f)g$ are bijections and $F_{xy}=\beta^{-1}\circ F_x\circ \alpha$ is bijective).

As a counter example for question two consider: $\Gamma=C_2$, $X$ the “complete” (in the sense of graph theory) groupoid consisting of two objects and four isomorphisms. The action of $\Gamma$ on $X$ is defined to swap the objects. Let $Y$ be the terminal groupoid consisting of a single isomorphism. $\Gamma$ acts trivially on $Y$. The unique morphism $X\to Y$ is a weak equivalence of $\Gamma$-groupoids. But there does not exist a $\Gamma$-morphism $Y\to X$.