So can I save and quit on emulator or not? It says saves are allowed but to my knowledge a save state is the only way to save on emulator isn't it?

That's referring to in-game saves, which are allowed and commonly used during the run. The rules explicitly say that emulator save states are banned.

1. Save and quitting is allowed but you must go back into game immediately, the main reason it being allowed would be rng manipulation and saving before a fight that isnt a guaranteed win so you can soft reset and refight when you lose

2. You can use normal saves on emulator, Menu -> Save -> Soft reset or hard reset -> Save will be there for loading, if its not there for you, please make sure youre only using an approved emulator (newest mGBA) and that the folder its in is writable.

For clarity given the above, you can't use savestates in a run (if you plan to submit it to these leaderboards), but you can use the in-game save function and reload that save if you want. If you want to take a break from the run however, the time will keep running, so you can't take a break mid run and remove that break time.