I don't know if I have a SSN... I think I have had one, but I am not sure.

This is going to sound crazy but I think I recall having a Social Security card, but I am now not sure! I remember doing a few things around my US citizenship like signing up for some draft/military thing (Selective Services) and probably the SSN at one point in my life, but once again I am not sure! This would have been many years ago, perhaps as far as 18 years ago. I did eventually find the Selective Services letter, but no trace of the SSN anywhere... I'm tempted to think I did this as well but now I have doubt, I kept everything US related in one place and there's no SSN there. 

I don't have a valid US passport either, I have one that expired many years back as well.

If I wanted to renew my passport can I do it if I don't have an SSN card? Is there a way of finding out if I have had a SSN?

How to get a new social security card:


https://www.ssa.gov/ssnumber/

There's really not a lot of reason to have the SSN if you don't plan on living in the US.  But have you never filed US income taxes?

Do you still have your expired US passport?  You would have to submit that to get a new one.