The only reason we'll look at pastes is if they are reported for violating our AUP, or if we get a DMCA complaint about them. We never snoop around what users paste, and it's none of our business.
If you create a public paste (public by default), your paste will show up for everybody in the menu on the right and on the archive pages. You can also create private pastes, these items will be invisible for others unless you share your paste link. These items can only be viewed by you when you are logged in and are therefore password protected. Search engines will only index public pastes.
That depends on you. You decide if you want your pastes to "expire". Setting an expiration date will automatically remove the paste after a certain amount of time.