The contained shape of the exhaust you see at sea level is being constrained by the atmosphere. It is expanding as much as it can, as it moves downwards.
As the rocket goes higher, the air pressure decreases and the exhaust plume expands. Watch a SpaceX Falcon 9 launch and it is really obvious. Actually, watch any sea level launch from a good camera position and it is really obvious. SpaceX just happens to be flying cameras and posting videos.
That is why the nozzles on upper stage engines are longer to prevent expansion as long as possible to maximize performance. An aerospike engine is designed to expand as per atmospheric pressure. I.e. Use the atmosphere as your nozzle, and it adjusts itself as the pressure changes.
Not sure why the exhaust appears to creep up the stage, but probably same reason.