if my bot is actually my cat actuating a switch for it to load a page, does it have to follow robots.txt?

robots.txt is more like a sign that asks certain people not to look at a bunch of other publicly visible signs.

One can't post a sign in public that tells people not to look at other publicly visible signs and expect the government to arrest or fine them for ignoring it.

robot != UA

What if I user curl to pipe web content to my mail so that I can read it in a quirky way? What if I write a Chrome extension to crawl a site? Where does w3m stands?

This is not a question of the tool (UA) but of the intent (mass crawling, indexing, mass-replicating stuff). robots.txt is made as hints for crawlers and the like, not optimistically ACL whether something is public or not.

robots.txt cannot change whether something is public, because it doesn't apply to humans.