Yes it does. You can include it in your deps array and not worry about it changing. For some cases, consider useEffectEvent

Yes — the AppRouterInstance returned by useRouter() from next/navigation is stable across renders. The two snippets in your question are equivalent.

Internally, the router is provided through a React Context (AppRouterContext) by <AppRouter /> in the App Router runtime. The context value is the same router object instance for the lifetime of the route — it does not get recreated on each render. Including it in a deps array will not cause your effect to re-run.

So this is safe and lint-clean:

useEffect(() => {
  router.prefetch("/x");
}, [router]);

Note: this is specific to next/navigation in the App Router. The legacy useRouter() from next/router returned a new wrapper object on each render, which is one of the reasons the App Router version was redesigned.