Let's say we have some set of argumentative rules. I will give one, right here:
So given that, is it possible to avoid all the fallacies they describe, and still come up with an incorrect statement? Why or why might that not be?
Yes. It is possible to construct a perfectly logical argument based on false premises.
Not merely the circular argument your list describes, but an assumption at the base of the argument that renders the logical conclusion incorrect.
This is usually employed to mask motivated reasoning (another fallacy).
Fallacy doesn’t exclude paradox. Leastwise, not in a context where logic and reason are your only tools. Zeno is logically correct in his argument against motion, it just doesn’t reflect a real world actuality. Nihilism isn’t logically wrong either, but neither is it “more right” than more optimistic schools of philosophy. Logically, there is no logical argument for using logic. At a certain level, you have to decide what you value and build up from there.
Okay, let's say we are past the concept of a paradox, and have resolved the general case for the paradox. Now what?