For the first sentence the correct version would be: 'My passport has expired' or 'My passport is past its expiry date.'
The problem with the sentence you used above is that it suggests that the expiry date has expired and not the passport. It is impossible that the expiry date itself could expire.
For the second sentence the correct version would be:
'No problem, just request to extend it. You can always do it even if it hasn't expired.'
'No problem, just request to extend it. You can always do it even if it is still in date.' (in date means not expired)
With the sentence above 'nevertheless' wouldn't work. Nevertheless can only be used to express a contrast. It is also nearly always used to start a new sentence:
'You can travel with your passport if it hasn't expired. Nevertheless, if it is within 6 months of expiring you should think about renewing it.'
In your sentence, you are not expressing a contrast, you are saying that you can extend your passport AND ALSO you can extend it if it is still in date.
With the last part of the sentence you should use 'hasn't expired' and not 'hasn't been expired'. 'Hasn't been expired' is passive, and means that someone else has expired the passport. In English, we would say that the passport expires on its own without someone else doing it.
I hope this helps! Let me know if you have any questions!