1. Trash bytes can exist in any text, if the game doesn't clear the data before writing the new text value. Not all GUI locations support editing trash bytes, but you can try via ctrl-clicking the textboxes to see which ones support it.
2. Correct, you can apply nicknames multiple times, from long to short.
3. Things can legally have trash bytes; depends from game to game. PKHeX defaults to the currently loaded save file when you are generating from a template, if there is no trainer in the trainers database that is better suited.
Japanese is just the first language. The dropdowns on the side are to apply Species names to the trash bytes for specific languages as an under-layer.