(1) is perhaps better rephrased as: (1') $\lambda\in\sigma_p$ (in words: $\lambda$ is an eigenvalue) because then both (1') and (3) are statements about spectral properties. Also, in more compact notation, we can rewrite: (3) $\lambda\in\sigma_d$

As you pointed out, (1') $\implies$ (2) is false, and so is (1) $\implies$ (3), for the same reason. (2) $\implies$ (1') and (2) $\implies$ (3) are manifestly absurd.

(3) $\implies$ (2) is true because $V\to 0$ implies that $\sigma_{\textrm{ess}}=[0,\infty)$. (3) $\implies$ (1) is obvious since $\sigma_d\subseteq\sigma_p$.

Finally, indeed nothing changes in higher dimensions. A radially symmetric version of the von Neumann-Wigner potentials still works as a counterexample to (1) $\implies\ldots$.

(1) is perhaps better rephrased as: (1') $\lambda\in\sigma_p$ (in words: $\lambda$ is an eigenvalue) because then both (1') and (3) are statements about spectral properties. Also, in more compact notation, we can rewrite: (3) $\lambda\in\sigma_d$

As you pointed out, (1') $\implies$ (2) is false, and so is (1') $\implies$ (3), for the same reason. (2) $\implies$ (1') and (2) $\implies$ (3) are manifestly absurd.

(3) $\implies$ (2) is true because $V\to 0$ implies that $\sigma_{\textrm{ess}}=[0,\infty)$. (3) $\implies$ (1') is obvious since $\sigma_d\subseteq\sigma_p$.

Finally, indeed nothing changes in higher dimensions. A radially symmetric version of the von Neumann-Wigner potential still works as a counterexample to (1') $\implies\ldots$.