an orientation of an object that may not have any dimension

The thing is, if it had no dimension, then there would be no way for it to have any orientation in some dimensions, it would have to be perfectly identical regarding all dimensions.

if the matter just outside a singularity rotates

It’s a bit more fun, because it would be normal for matter to orbit around before falling in, but “frame dragging” means that not just matter, but also light outside it rotates with the black hole, and time gets stretched.