This is true. The functor $RHom(\mathcal{O}_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z_n \to X$. Where $Z_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In the limit we get a kind of $j^!$ too only now for the inclusion of the formal scheme $\widehat X_Z$ (formal neighborhood of $Z$ in $X$) into $X$. When $i:U \to X$ is the inclusion of the complement we have an exact triangle in the derived category

$$0 \to j_*j^! \to Id \to i_*i^* \to 0$$

So for the case of $X$ abelian we indeed have that local cohomology is identified with the shifted version of the pull push along $U \to X$.

This is true. The functor $RHom(\mathcal{O}_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z_n \to X$. Where $Z_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In the limit we get a kind of $j^!$ too only now for the inclusion of the formal scheme $\widehat X_Z$ (formal neighborhood of $Z$ in $X$) into $X$. When $i:U \to X$ is the inclusion of the complement we have an exact triangle in the derived category

$$0 \to j_*j^! \to Id \to i_*i^* \to 0$$

So for the case of $X$ affine we indeed have that local cohomology is identified with the shifted version of the pull push along $U \to X$.

This is true. The functor $RHom(\mathcal{O}_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z_n \to X$. Where $Z_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In the limit we get a kind of $j^!$ too only now for the inclusion of the formal scheme $\widehat X_Z$ (formal neighborhood of $Z$ in $X$) into $X$. When $i:U \to X$ is the inclusion of the complement we have an exact triangle

$$0 \to j_*j^! \to Id \to i_*i^* \to 0$$

So for the case of $X$ abelian we indeed have that local cohomology is identified with the shifted versuion of the pushforward along $U \to X$.

This is true. The functor $RHom(\mathcal{O}_X/\mathcal{I}^{n+1},-)$ is in fact a different name for $j^!$ when $j: Z_n \to X$. Where $Z_n$ is the $n$-truncated formal neighborhood of $Z$ in $X$. In the limit we get a kind of $j^!$ too only now for the inclusion of the formal scheme $\widehat X_Z$ (formal neighborhood of $Z$ in $X$) into $X$. When $i:U \to X$ is the inclusion of the complement we have an exact triangle in the derived category

$$0 \to j_*j^! \to Id \to i_*i^* \to 0$$

So for the case of $X$ abelian we indeed have that local cohomology is identified with the shifted version of the pull push along $U \to X$.