Just as the question asks.
Will the Roadster get anywhere near, or be in any planned position to capture images of the Earth and Moon in the same field of view?
A sort of "family photo" :)
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We know that the moon is visible in the Roadster footage, so a photo with the Earth and the moon is possible. Here the moon can be seen in the lower left of the frame:
The Earth was visible in-frame about one minute before this frame. As the Roadster orbits the Earth the three objects should line up for a shot, especially with the wide-angle lens on the front camera. There are hours of footage, if I see a shot with both in-frame I'll update this answer with the definite YES.
EDIT: This was a total nerd snipe. The best "family photo" I could find was the reflection of the Earth on the bodywork with the moon in the background. Enjoy!
I believe the battery is already dead. Elon Musk estimated that it would die in 12 hours. The live feed stopped after four hours. Even if the battery is not dead, it would certainly be dead by the time the Roadster would be far enough to make this shot.
The 12 hour battery life expectancy comes from here in the press conference: