Pan & zoom from still image with green screen
If you want parallax, you can do it without a video, a still image of the phone could work, and would make things a lot easier and faster.

If you want a video you need to add markers to the green screen so the program can track them.

You could also have just simple moves, for example, your object only rotating, so you do not need to track the background. Some simulated moves for the new background on post could be enough.

For the DOF you can potentially add one problem if you want the DOF on your product. It is better if you film your product very sharply, so the green screen cuts nicely. The DOF would be a nightmare.

Then using a real 3D program you can add your video on a layer, track the green screen, generate a 3D scene, and replace the green screen with another plane with a photo.

I would use Blender for that. But if you want simple moves and not a complex 3D tracking you could go away with HitFilm, which has 3D planes too.