About false theorems: we still don't have any. The examples
given were all correct results with incorrect or incomplete proofs.
The Four Color Theorem, the Hard Lefschetz Theorem, and Dehn's
Lemma all turned out to be true. Not only that, the erroneous
proofs were noticed by human mathematicians, not by automated
reasoning systems. I have never encountered a false theorem
that was used as the foundation for other theorems, with
disastrous results. And I don't think anybody else has, either.
There are many imperfections in the mathematical literature,
and some incomplete proofs, but I don't think there are any
substantive errors that affect the integrity of mathematics.
Isn't this what people call a selection effect? We don't
remember false proofs of falsehoods!