In the sense that they get to take the profits home with them.
Can they sell or transfer this ownership to someone else?
Yes, through sale of stock in member banks. See post 24, above
Can they file for bankruptcy?
Non Sequitor. Why would you declare bankruptcy when you can create the worlds reserve currency out of thin air?
This does not sound like fictional ownership to me. If you're still struggling with the concept, go ask Mr. Spock!
Since you are apparently arguing that ownership shares (and thus potential control if enough are owned) of the Federal Reserve can be purchased by private individuals, please tell me where I can buy them. Are they traded on the stock exchange? Again, please give me the contact information of the person or entity that sells them.