This is a broad question, but you may find it helpful to read The Existential Risk of Math Errors. It suggests a certain robustness of the mathematical edifice, which I actually think extends to the natural sciences as a whole. (Newtonian mechanics is "wrong" in a fundamental sense, but neither the development of relativistic mechanics nor the discovery of quantum mechanics has not caused the collapse of classical mechanics.)
This quote in particular from Gian-Carlo Rota bears on your points 1 and 2:
When the Germans were planning to publish Hilbert’s collected papers and to present him with a set on the occasion of one of his later birthdays, they realized that they could not publish the papers in their original versions because they were full of errors, some of them quite serious. Thereupon they hired a young unemployed mathematician, Olga Taussky-Todd, to go over Hilbert’s papers and correct all mistakes. Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties. At last, on Hilbert’s birthday, a freshly printed set of Hilbert’s collected papers was presented to the Geheimrat. Hilbert leafed through them carefully and did not notice anything.
