Does anyone know if typical/common worms and worm gears self-locking (plastic or metal)? Are the self-locking ones specially designed (with specific angles) in order to be self-locking? By self-locking, I mean that the worm gear can't turn the worm but the worm can turn the worm gear. Are they considered "special" worm drives, and thus cost more to make (and thus, less commonly used)?
The vast majority of worm gears are self locking naturally, it wouldn't be considered a special feature, just something you should confirm in your final design. Generally any single start worm screw will be self locking, there are multi-start worm screws that may not be.
If you are doing something where backdriving would be hazardous (not just risk minor damage to machinery) you should probably have an independent brake and not rely on the gear mechanism to hold a sustained load.