Nature of time. First things first.
"If everything in the universe stopped moving in an absolute sense..."
Point: "Since motion in the universe can only be defined in relation to arbitrary reference points, per Relativity Theory, it is not possible to stop moving in an 'absolute' sense".
Counter-point: "But if every reference point appeared to be motionless in relation to every other reference point, that would effectively be cessation of motion in an 'absolute' sense."
Point: "Not necessarily. Every reference point could be apparently motionless in relation to every other, but the universe as a whole might be rotating. Alternatively, the apparent motionlessness might be only a special case of relative motion, just as the number 0 is a special case of 'number', in that it has no positive or negative value."
Counterpoint: "But how would we know that the universe is rotating? After all, the universe is by definition 'everything'. By what reference point could it possibly be rotating?"
Point: "Our definition of 'universe' is itself framed within the limitations of our own understanding. The fact is, we simply have no firm conception whether there might be anything 'outside' of the known universe, in either space or time. The question whether it might be rotating in respect of some unknown (and perhaps, to us, inconceivable) reference point therefore remains open."
Counterpoint: "And as to the apparent absolute motionlessness being merely a special case of 'motion', how could we possibly establish that?"
Point: "We can't. But as a matter of scientific logic, in the absence of any internal logical contradiction, the hypothesis cannot be excluded."
Counterpoint: "So the question whether everything in the universe could stop moving in an absolute sense is unanswerable?"
Point: "I'm afraid so."