Multiple sign-in is a per-browser session, meaning that just because you signed into 2 accounts in one browser, they both won't necessarily be accessible in another browser without explicitly signing into the accounts again.

Signing out / logging out is not accomplished by just closing the tab / browser - you have to click on the account name in the top-right corner and then click sign/log out (I can't remember the exact wording).

That should drop you back to the google sign-in page.

The first account you sign in to will be considered the 'primary' account and things like docs, YouTube, search history, and any other service that doesn't support multiple sign-in will be associated with the primary account.

The multi-sign in feature is permanent in that once you enable it, you can always sign in to more than one google account (as long as google provides that feature). It is not limited to yours and your fathers accounts - you will be able to log into any google account that has multi-sign in enabled.