"If P, then Q. P. Therefore, Not-Q." is just as basic and elemental an error as "If P, then Q. Q. Therefore, P." is.

I'm not sure I'd grant that. The second can be sneaky, in that you can encounter countless arguments of that form with true premises and a true conclusion. In the first example, on the other hand, true premises guarantee that the conclusion is false.

I'm not sure if there's a word for the latter category, but there probably should be. "The conjunction of the premises is inconsistent with the conclusion" is not nearly as awesome as, say, "Antivalid"