Using Google's new code search thingy, I see that ffmpeg has a .STR file reader. It's GPL too so we can rip directly from this to play the movies directly from the disk.
Code segment is Here
ffmpeg also has the ability to read PSX ADPCM data too, so that solves some video/audio issues....
I'm sure at one point I was able to play FF7 movies, but when I tried it recently it didn't work. STR and DAT are not standardised in any way, so I assume they just work together with the SDK version used to build the game - earlier or later SDKs have small tweaks or fixes and may not be compatible.
So far I was too lazy to go through the archives to find a version that works, or hack the STR reader until it works with the data files used in FF7.
Luckily the actual M-JPEG decompression is fixed, as they used the MDEC hardware. And some bits of the STR format are limited by the CD frame format used. But they seem to use a Huffman compression scheme in-between which has changed in the latest version of ffmpeg, probably to support newer games.