Everything can't rotate because there's nothing else for it to rotate relative to.
Not only that, but if it was rotating, it would have an obvious centre and we would see everything orbiting it.
I don't think so. Not necessarily.
Consider a universe - the whole shebang, everything, no external reference frame. Separate it into two regions, of (approximately) equal size ; set one part rotating clockwise relative to the other, and the second part rotating anticlockwise relative to the first.
You now have a universe where everything (except that on the rotation axes) is rotating, but the net angular momentum is zero.
But yes, if (if) your local "observable" universe included one or other of the rotation axes, then you should have something to look at. If the axes are outside your observable section of the universe, maybe you'd be able to tell, maybe not. My maths isn't up to saying for sure, either way.