Thank you for the suggestion, kmielczarcyk. I followed your instructions, but nothing changed.
I actually do not fully understand your suggestion. Doesn't Keychain just store and insert the user name & password into the correct app/website? As I stated in my original post, the username & password were already being populated. It's just that they were not bringing me right to being logged on. What I'm not seeing on these pages is the checkbox "stay signed in". Could that be the issue? And what could be the solution? And, I was just using Facebook & LinkedIn as examples. The issue occurs with many websites, but only with Opera.