None of your "
extraordinary claims" has any
empirical content. You're right, there are no
empirical proofs for such claims.
Contrast this with a claim like the existence of horned pink dragons or ESP. These are claims with empirical substance (horned pink dragons are dangerous pests, who will eat my homework if they get a chance; ESP lets me read your mind unless you wrap tin foil around your head). These claims are similar to claims of the existence of the moon and a human ability to do maths; they're just a little more outlandish. In other words, we'd like some fairly impressive proofs of them. Strong enough that Occam's razor will make us conclude that they're more likely to be true than that the "proofs" are somehow wrong.
Invisible pink unicorns fall into the first (unempirical) category, unless I claim some real effect of their existence, which is unexplainable by any other means (including transparent blue unicorns). Of course, I shall be very careful not to make such claims if I wish to retain my credibility.
None of your 3 points has any empirical content that I can glean, it is true. So, by analogy, you'd reason it is impossible e.g. to rule out the existence of God by purely empirical means. This is indeed true. But any claim of empirical proof of God's existence will have to be extraordinary. There are other paths available to you than the scientific one; I imagine many of the religious scientists who exist take these paths. But claiming "scientific proof of God's existence" (or your favourite parapsychological phenomenom, or the like) will require scientific proof.
You don't have to play in Science's playground. But if you do decide to play there, you have to play by the rules.