Is there any theory that states that the universe is moving? By that, I mean the universe is moving relative to a point in a different space containing the universe. By analogy, the universe would then be like a moving train with the planets inside of it its passengers. Is there any theory or idea that expresses this possibility?
Yes, brane collision in M-theory, where our universe is pictured as the collision 'line' between higher dimensional surfaces.
Compare to angular momentum, which in non-higher dimensional theories is widely considered to only exist relative to the restframe of the whole universe (ie, net zero over all). Edit: To clarify, in this picture the universe spinning would not create any change, in fact it would be meaningless, and similarly any translational momentum/velocity of the universe as a whole (in an open, 'saddle shaped' cosmological spacetime, this would seem possible).
Our star is slowly orbiting the Milky Way galaxy's centre, so our solarsystem's planetary orbits are spirals not circles, relative to the galaxy as a whole.
Relativity favours no specific frame of reference, it only provides a recipe to convert between different inertial (non-accelerating) rest frames.
You may find this discussion interesting: Can space be distorted without things that occupy the space being distorted?