A hook will rust away in a fish, but it may take a while, especially if the hook is plated or made of thick metal. But fish's stomachs are pretty tough. They can stand up to the spines on little fish like bluegill. I've caught lots of fish which have hooks embedded in their jaws, and I've cleaned quite a few fish which have grown-over hooks in their guts. So cutting off a swallowed hook is not really a big deal. But if you worry about it, make sure you use thin wire, non-plated hooks.
And no, fish's nervous systems are not especially similar to ours. I've actually seen the same fish caught twice within an hour (we were tagging small halibut, so we know they were the same fish and when they were caught), so hooking a fish doesn't bother it much at all. You also see this in the big fish tank bass fishing exhibits you see at fishing shows. And of course, fish have lateral lines, we don't. They are very different creatures.
Finally, I've caught a number of fish which were never hooked: lingcod which have grabbed a smaller hooked fish, or wahoo which are simply holding the middle of a metal jig, with the hook swinging free. These fish pulled the same as fish which were hooked, and held on until they were gaffed or netted. All this tells me is that they're fighting the pull of the fishing line, not the "sensation" of the hook, and they were not bright enough to let go when it became obvious they were in danger.