No it wouldn't. It would be Doppler shifted. If the craft is moving away, it would slow the video rate some small amount. If the craft was moving toward you it would speed up the frame rate a small amount.
The higher the velocity relative to the receiving end, the greater would be the change in the frame rate.
No. Because the video recording is being taken relative to the crew and being sent. The transmission itself is not relative to the ship. You would see a normal video.
If you were OBSERVING the ship itself it would be doppler shifted. But you’re watching a video from inside the vehicle travelling at light speed.