When Beanie moves the ship, the view of the bottom shows the shuttle, It was not attached to the ship at that time.
The ship supposedly catches up with radio waves broadcast from Earth in the past. However, their objective was only 18.7 light years away. Especially since they had only begun the trip, they should not have received any broadcasts older than a few weeks or months old. The broadcasts they received range from 1927 to 1987, which should have been 101-161 light years from Earth by the year 2088, more than five times the distance to Demeter. Also, because light cannot vary speed, they would not have caught up to all the broadcasts at once. The broadcasts would have gradually gotten older as they traveled further from Earth.
(Additional note: There is no reason to suspect that the intercepted radio signals were original broadcasts; although that is the aside made in the movie. These could be rebroadcasted (reruns) programs from Earth on the same day in the same spatial direction.)