> How would the input and output devices work without drivers? Does "no classical software at all" mean that the drivers are somehow within the neural net?

Why do you single out drivers? If the whole I/O behaviour of the computer would be performed by the NN, why not drivers?

> Wouldn't it require observing countless inputs and outputs for a normal computer?

Yes and that bit sounds to be relatively easy to automate.

> How would the neural net being trained distinguish between good, wanted input/output combinations versus bad, error input/output combinations?

I guess you would put that information into the dataset. In supervised learning it's called labels.