The nail battens must go across the facade panel, so with vertical cladding, the nail battens are horizontal, and with horizontal cladding, the nail battens are vertical. In the past, it was quite common not to have vertical nail battens and instead nail directly into the wall studs with horizontal cladding, but it is probably less common today as people prefer to build with an air gap.
Yes, of course, you may and must screw/nail through the weather barrier.