Does the Oort cloud rotate? Or is it static?
The Oort Cloud certainly moves and most probably revolves around the Sun.
Does the Oort cloud rotate? Excellent question!
Ok - first thing to know is that the Oort cloud is thought to exist but hasn't been proven to exist - so that's the first thing to keep in mind.
If in fact the Cloud does exist, the next question is - does it move? The answer is almost certainly yes. Keep in mind that in deep space there are 2 factors working out there:
gravity - so the pull of one object by another, whether it's elements of the Cloud on other elements, the Sun and other planets pulling on it, and the gravitational effects of neighbouring stars, and
the lack of friction to stop the movement of the objects being pulled.
So then the question becomes - does the Oort Cloud revolve around the Sun (to rotate means to spin on it's own axis, such as the Earth as it spins and causes the Sun to rise and set. To revolve is to move around a different object, such as the Earth as it moves around the Sun and creates seasons)?
Again, the answer is almost certainly yes. Keep in mind that it's the gravitational pull of the Sun on these objects that keeps them from simply floating off into deep space, so there is a constant pull from the Sun. But they also aren't falling into the Sun, and so the motion of the objects of the Cloud, and therefore the Cloud itself, would orbit the Sun.
As an example we are more familiar with, let's look to Saturn's rings. The rings themselves revolve around Saturn and the speed is dependent on how far away the ring is from Saturn's surface.