is there even a correct password for the vault?
Depends what day
Depends on what path you want, pay attention to what is said during the days and it will hint the passwords
To my understanding yes (to actually open the vault not just use it) but it is currently unobtainable.
Passwords on Day 1+2 cannot be solved so far.
Day3 is a bonus passwort, not relavant to the story.
Day 4-8: Can be solved, AFTER you experienced the bad ending at least once.