You are confusing and mixing up a lot of things. Let me start with mathematics. Mathematics is a very special case and very different from other sciences. It is completely made up by humans and has nothing to do with our world. We use mathematics in other sciences to describe our world but on its own mathematics has nothing to say about our world. You are also kind of misinterpreting Gödel's incompleteness theorems. The first incompleteness theorem states that a system of axioms is either inconsistent or there are true statements that can not be proved from the axioms. Intuitively - but probably wrong in a strict sense - one could say that the number of true statements is just larger than the number of proofs, see Cantor's diagonal argument. The second incompleteness theorem states the consistency of a system of axioms is one of the statements you can not prove by the system of axioms itself. But while you can not prove the consistency of a system within itself you can use a different system and so for example Gentzen's consistency proof proves the consistency of first order arithmetics within primitive recursive arithmetics. I also don't understand why you think of the axiom of choice as a debacle. The axiom of choice is independent of Zermelo-Fraenkel set theory and it is up to you whether you want to accept and use it as an axiom or not. And also Russell's paradox has been resolved long ago when mathematicians realized that not all collections of objects are sets.

Verifiability in the context of science means you can reproduce and verify results, i.e. when your quantum physics book says that electrons hitting a barrier with two slits will cause an interference pattern on a screen you can just do the experiment yourself and verify that it is true what your book says, no need to blindly believe anything, at least in principle. As far as I can tell religions lack this property.

You are also confusing natural sciences like physics and engineering disciplines like architecture. In physics a theory gets replaced by a new theory when discrepancies between theory and reality are found - Newtonian gravity does not exactly explain all gravitational phenomena and therefore got replaced by the general theory of relativity. Engineering disciplines on the other hand are not concerned with understanding the world and just use whatever fits their needs - why use the complex theory of general relativity to construct a building when Newtonian gravity is a simple and good enough approximation? By the way, black body radiation lead to the quantization of light, not to the special or general theory of relativity and there are a few more misconceptions in there but I will not go into them.

Predictibility means that a new theory predicts something nobody has observed before and you can look for it. If you actually find, it is a strong hint that the theory might be up to something. In 1928 Paul Dirac formulated a relativistic quantum theory for the electron and realized that this theory predicted an antiparticle, the positron, and four years later it was actually discovered. In 1916 Einstein completed the general theory of relativity and predicted the deflection of light by heavy masses, in 1919 this deflection was observed during a solar eclipse. Sitter precession and Lense–Thirring precession were predicted in 1918 but it took until 2004 that we were able to build Gravity Probe B and launch it into space to test the prediction. The Higgs particle was also predicted 50 years before it was discovered in 2013. Again as far as I can tell religions lack this property and your examples don't really fit into this category.

(I will answer to the remaining points tomorrow but right now I am running out of time.)