No, we know for a fact that your head will live after it's chopped off for a bit. And the thing I was mentioning about the eyes winking was a specific incident where a scientist was executed (well, philosopher), and asked his friends to see how long he was alive for. Not a mob doing it. There's the mob-hand accounts of it too, but I wasn't referring to those. As for the shock, yes, that's a possibility, but not a guarantee. And the fall will absolutely NOT knock you out. It's no where near far enough to cause severe trauma unless you're in some new guillotine execution style where they aren't letting it fall into a basket with cloth. You're probably awake for at least some portion of it, and then unconscious for a good portion of it, and then dead. It might even take up to 10 minutes to die in some cases. But you're more likely to die in around 4, and be awake for at least 10 seconds. Which is PLENTY for me to never want to go like that. Hell, even the trauma that would put you into shock, you don't want to experience THAT. Just because you're out doesn't mean you didn't feel what put you out!
As for hanging, no, absolutely not. Hanging, properly done, is worse than decapitation, simply because you'll be alive for longer. Not a lot, but some. And hanging, improperly done, is many times worse, because you might not even die. from the hanging, you'll suffocate, and that's just the worst way. Not to mention that there's a risk of this happening, whereas with a guillotine, you're pretty damn sure it's going to work.
As for the chemical cocktail, it's probably the best. Especially if you're knocked out first with a gas. There's still the chance of it not working, which is awful, but it's probably the best. You can have a chemical that literally just eliminates all pain. Boom, done. NO pain. Just put enough in. The issue is that everyone reacts differently to different things, but all in all, chemical is probably the way.
Now, the only way to guarantee no pain however, is to instantaneously destroy the brain. But good luck with that! Try finding a meteor on a collision course with your head or something.