It's not exact in general. Take $\mathcal{A}$ to be the category of finitely generated abelian groups, so that $\underline{\mathcal{A}}$ is the category of finite abelian groups and $\Omega=0$. Take the short exact sequence $$0\rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2$$ and $A=\mathbb{Z}/2$. Then the long sequence is $$\mathbb{Z}/2\stackrel{1}\rightarrow \mathbb{Z}/2\stackrel{0}\rightarrow \mathbb{Z}/2\rightarrow 0\rightarrow \cdots,$$ which is not exact.

It's not exact in general. Take $\mathcal{A}$ to be the category of finitely generated abelian groups, so that $\underline{\mathcal{A}}$ is the category of finite abelian groups and $\Omega=0$. Take the short exact sequence $$0\rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2\rightarrow 0$$ and $A=\mathbb{Z}/2$. Then the long sequence is $$\mathbb{Z}/2\stackrel{1}\rightarrow \mathbb{Z}/2\stackrel{0}\rightarrow \mathbb{Z}/2\rightarrow 0\rightarrow \cdots,$$ which is not exact.

It's not exact in general. Take $\mathcal{A}$ to be the category of finitely generated abelian groups, so that $\underline{\mathcal{A}}$ is the category of finite abelian groups and $\Omega=0$. Take the short exact sequence $$0\rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2$$ and $A=\mathbb{Z}/2$. Then the long sequence is $$\mathbb{Z}/2\stackrel{1}\rightarrow \mathbb{Z}/2\stackrel{0}\rightarrow \mathbb{Z}/2\stackrel{1}\rightarrow 0\rightarrow \cdots,$$ which is not exact.

It's not exact in general. Take $\mathcal{A}$ to be the category of finitely generated abelian groups, so that $\underline{\mathcal{A}}$ is the category of finite abelian groups and $\Omega=0$. Take the short exact sequence $$0\rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/4 \rightarrow \mathbb{Z}/2$$ and $A=\mathbb{Z}/2$. Then the long sequence is $$\mathbb{Z}/2\stackrel{1}\rightarrow \mathbb{Z}/2\stackrel{0}\rightarrow \mathbb{Z}/2\rightarrow 0\rightarrow \cdots,$$ which is not exact.