If not mistaken pictures need to be downloaded before you power off.
Not 100% on that as I never look at them from app just after dumping my card in Mac.
That's correct. The Fly app saves low-res videos on phone, but hi-res videos (much better) and
all of the photos are saved on the memory card in the drone. They are NOT saved on the phone unless/until you download them to the phone manually.
You can "download" the photos to your phone - while the drone is connected to the controller and the DJI Fly app is still running - by backing out of FPV mode (when you can see the live video feed coming from the drone) and clicking on "Album" (lower left-hand corner of the screen)

and then clicking the download icon. At that point, in the upper-right-hand corner of the app is an icon that allows you to choose multiple files at once. If you select multiple images and then click "Download" on the bottom of the small window, the images will be copied to your phone, in the DJI Fly folder. You can then view them on the phone, as you would any other image...
Note: Copying images to your phone does not erase them - they're still on the sd card and can later be moved to your computer.
The low-res videos saved on the phone might have problems if, during your flight, you partially lost connection - there might be gaps or pixilation in the low-res videos - that don't exist in the better hi-res videos you'll find on the micro-sd card. Always use the hi-res video for important projects, but if you're just uploading for fun on facebook or something, you can use the lo-res video. Chances are, no one will notice the lower resolution...
Enjoy