Parts of my characters’ costumes vanish or go flat.

In some situations, parts of characters or other models will vanish or flatten. This has particularly been observed with cel-shading turned on on some Macs.

On the front screen, go to the 'Settings' menu, and on the 'Graphics' tab, disable Hardware Skinning.
Try loading your scene again - hopefully it now works.

Otherwise, you may need to disabled Cel-Shading for your movie (if it was enabled), or disable Shaders (from the 'Graphics' settings again).