Good afternoon. Your wife will require a sponsor unless she qualifies as an extraordinary teacher under a special category reserved for those who have risen to the top. Then she could file a petition on her own, however it is a very high standard.
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Dear Sir or Madam: You will not necessarily need a passport to apply for a "green card", however, you will obviously need a document such as a passport or reentry permit to travel internationally.
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Dear Sir or Madam: I would agree, you should take Form I-20 and perhaps even a copy of your application for OPT and employment authorization and school transcripts in a separate envelope to be presented only upon request to be on the safe side. This way you will be able to show that you are still within the authorized period of your F-1 visa.
Dear Sir, If your US citizen wife were to divorce you prior to the second interview, you would then have to send the proof of the divorce and file a waiver with the USCIS (immigration) in an effort to maintain your residence. You would need to prove that the marriage was entered into in good faith, so I would recommend strongly that you begin to gather as much proof as possible of the validity of the marriage. This will be a factual presentation to the immigration officer as if you were...