Hi,

• If I tick the auto login box in the edit profile page and save
  
• log out

If, you are logging out, you also disable the auto login option.

• If I tick or untick the auto login box in the edit profile page and save
  
• close the browser (without logging out)

How long do you wait between closing and re-opening the browser? The cookie should be valid for a longer time (depending on the forum settings).

/Micha