1. Don't double click! You are telling Moodle to do things twice. It is a button - it should only be clicked once.
I would guess this is causing the majority of your issues. The backup and import is notorious for having issues if there are problems with the quiz bank which if you have duplicate questions you have issues..
Fix it ideas.
Delete the duplicate questions from your quiz bank. If they have never been added to a quiz, you then want to go and delete them from the database. Refer to Tim Hunt's database diagram in the quiz documentation to see where else they might be referenced...make sure you keep the quiz id numbers handy. Be careful with this...
It looks like your database connection might be timing out or your backup might be corrupted. Check your mysql settings and increase the timeouts. I suspect that the timeout however might be being caused by your question bank. After fixing the duplicate questions, try recreating the backup and try again.