Is our universe stationary ?

There is no possible experiment be performed by observers on an object that can distinguish whether that object is stationary or travelling at a constant velocity. That's what relativity tells us. But, in fact, it's more profound than that - things aren't either moving or not; we can only decide which they are doing if we pick a fixed point.So, in answer to your question: the universe can be both, and either, moving and stationary; but since both outlooks are equally valid we may as well pick the conceptually simplest and consider the universe to be stationary.