Assuming that the speed of the space craft remains much less than the speed of light, the time within video will be stretched as distance from earth increases. So motion as viewed from the broadcast will be slowed.
No it won’t! The transmission itself isn’t travelling away from the viewer at the speed of light. Only the ship is. The transmission is travelling in the opposite direction relative to every position of where the ship IS when broadcasting. The waves don’t travel with the ship. They are INDEPENDENT of it. Thus not relative to the ship, but to the viewer.
No. The relative rates of time passage are strictly dependent on relative speed and do not depend on distance whatsoever. A message sent from a craft traveling at 1/2 light speed will be slowed by the same amount ("the time within the video") regardless of whether the source is a foot away or a light year. This holds true no matter what the difference in relative speed is; it simply isn't noticeable when the velocities are very close to the same value. If we could actually distinguish it, the time rate of the guy walking toward the front of a moving bus is slower than that of the guy sitting in a seat, and both are slower, but not equally so, than that of a guy standing on the street as the bus goes by.