Random noise is supposed to change for each frame. Noise generated by screen coordinates only will look like it's sliding over geometry as the camera or the geometry moves, and generally looks shit. It's also impossible to properly filter out in post using motion vectors or optical flow techniques. If you're using Fusion or Nuke, I can probably point you in the right direction. For After Effects, your options are more limited.
Corona looks like it's doing things right.
The problem with the noisy DOF is something that will need to be solved with either noise reduction in post or better adaptive sampling in Corona.