$2$-extendibility reflects $2^\kappa$-supercompactness. It suffices to show that any $2$-extendible $\kappa$ is $2^\kappa$-supercompact. Then if $\mathcal U$ is a normal fine $\kappa$-complete ultrafilter on $P_\kappa(P(\kappa))$ and $j : V_{\kappa+2}\to V_{\kappa_* + 2}$ witnesses 2-extendibility, $\mathcal U\in V_{\kappa_* + 2}$, and so by the usual reflection argument, $\kappa$ is a limit of $\bar \kappa$ that are $2^{\bar \kappa}$-supercompact.
To demystify Kanamori's result, we show something stronger:
Proposition: If $\kappa$ is $(\eta+1)$-extendible then $\kappa$ is $|V_{\kappa+\eta}|$-supercompact unless $\eta$ is an infinite limit ordinal with $\text{cf}(\eta)< \kappa$.
The case $\eta = 1$ is what we need above. The cofinality restriction is necessary: if $\kappa$ is $|V_{\kappa+\gamma}|$-supercompact where $\gamma$ is a limit ordinal with $\text{cf}(\gamma)< \kappa$, then $\kappa$ is $(\gamma+1)$-extendible but not $|V_{\kappa+\gamma}|$-supercompact in the ultrapower by any $0$-order normal fine $\kappa$-complete ultrafilter on $P_\kappa(|V_{\kappa+\gamma}|)$.
Proof of Proposition. Suppose $j : V_{\kappa+\eta+1} \to V_{\kappa_*+\eta_*+1}$ witnesses $(\eta+1)$-extendibility. We claim that $j$ extends uniquely to an elementary $\hat j : H_{|V_{\kappa+\eta}|^+}\to H_{|V_{\kappa_*+\eta_*}|^+}$. Grant this for now. The cofinality restriction on $\eta$ implies that $|P_\kappa(|V_{\kappa+\eta}|)| = |V_{\kappa+\eta}|$. Therefore $P_\kappa(|V_{\kappa+\eta}|)\in H_{|V_{\kappa+\eta}|^+}$. Hence every subset of $P_\kappa(|V_{\kappa+\eta}|)$ and every function $P_\kappa(|V_{\kappa+\eta}|)\to |V_{\kappa+\eta}|$ are in $H_{|V_{\kappa+\eta}|^+}$. Moreover $\hat j[|V_{\kappa+\eta}|]\in H_{|V_{\kappa_*+\eta_*}|^+}$. So we can set $$\mathcal U = \{X\subseteq P_\kappa(|V_{\kappa+\eta}|) : \hat j[|V_{\kappa+\eta}|]\in \hat j(X)\}$$
and this is a normal fine $\kappa$-complete ultrafilter.
To obtain this extension $\hat j$, use the standard coding fact that for any infinite $\alpha$, there is (uniformly in $\alpha$) a model theoretic interpretation $f_\alpha : V_{\alpha+1}\to H_{|V_\alpha|^+}$. Essentially $f_\alpha(x) = s$ if $x\subseteq V_\alpha$ codes $(E,p)$ where $E$ is a wellfounded extensional relation on a set $A\subseteq V_\alpha$ with $p\in A$, and $\pi(p) = s$ where $\pi : (A,E)\to (M,\in)$ is the transitive collapse of $E$.
Let $f = f_{\kappa+\eta}$ and $f_* = f_{\kappa_*+\eta_*}$. One then sets $\hat j(s) = f_*(j(x))$ for any $x\in f^{-1}(\{s\})$ and checks that $\hat j$ is well-defined and elementary.
Second question. The question is a bit vague: it depends what you mean by "stronger than" and what you mean by "known $\eta$." But anyway the answer is probably no. For example if $\kappa$ is supercompact, then $V_\kappa$ satisfies that for any $\delta$ there is some $\bar \kappa$ that is $\delta$-extendible. Moreover if $\kappa$ is supercompact and $\eta > \kappa$, then there are arbitrarily closed $\kappa$-complete ultrapowers of $V$ in which $\kappa$ is $\eta$-extendible. In particular, if $\varphi(x,y)$ is a $\Sigma_2$ formula with parameters in $V_\kappa$ and there is some $\alpha > \kappa$ such that $\varphi(\kappa,\alpha)$ holds, then $\kappa$ is a limit of cardinals $\bar \kappa$ that are $\bar \alpha$-extendible for some $\bar \alpha < \kappa$ such that $\varphi(\bar \kappa,\bar \alpha)$ holds.
