No, that's not what we should take away from this. Bayes' theorem is a mathematical theorem, after all, not a theory that you can falsify or improve upon. If you plug the correct inputs into the formula, it is guaranteed that you will get the right inputs back out.

So if you are getting a more useful result by modifying the formula, that means that your inputs are wrong. In this case, it can be helpful to make "modifications" to Bayes' theorem, but really what you're doing is adjusting the inputs.

No. Peter Norvig found a particular instance where a formula motivated by Bayes' theorem worked even better when the formula was changed. The altered formula doesn't have a theoretical basis, but it seems to work.