Well, sort of - but we use "if" in day-to-day as a different thing to the material implication in logic. I've yet to run into a person using 'if and only if' in a way that doesn't map to the material equivalence, though, but I'm sure it must happen

Actually, this wikipedia article for has got a specific section about that https://en.wikipedia.org/wiki/Material_conditional

Edit: I also just learned abou the 'Wason selection task' from that link. Cool stuff. https://en.wikipedia.org/wiki/Wason_selection_task

yes, "if" statments in proofs are the "p->q" forms from logic, if you have p you also have q (therefore if no q then no p). If you imagine the set of all possible logical propositions as nodes in a graph, p->q tells you that q is reachable from p. If you prove that q is not reachable from whatever axioms you have (i.e. false), then p is false too by necessity, because otherwise q would be reachable and we know that's not the case. But if all you have is that q is reachable, that's not enough to assert that p is also reachable. Because you have no reason to believe that the road from p to q is the only one, it might very well be that p is false but q is reachable from another statment s or something.

"if and only if" is "p->q and q->p" (also equivalent to "p->q and not(p)->not(q), the more intuitive sense of the phrase), it basically establishes equivalence: any proof that requires p also requires q and vice versa, any proof that guarantees p also guarantees q and vice versa.

I think so. The loosey-goosey language of "if" is usually (would need to actually check this) interpreted as an implication relationship

"B if A" or "If A, then B" interpreted as

A -> B

The fallacy would then be to do the following

A -> B

B

------------

A

On the other hand, iff is interpreted as A <=> B

i.e. A -> B and B -> A

So you can get A or B if you have B or A respectively.