You are going to be in removal proceedings. You need to hire an experienced attorney to review your documents and see the best way to avoid removal. Make sure to notify the USCIS of your address if you move because your removal documents will be mailed to the last address they have.
You cannot transfer the petition to new organization; you will need to re-file under the new entity. Make sure the new entity meets all the requirements for R-1 petitioner and you are in valid status when you refile.
Yes, as a United States citizen who obtained your green card through the Diversity Visa lottery, you can file immigration petition for your parents. The place you live determines where to file; and your parents' immigration background can determine which forms to file, when to file, and whether your parents can get status in the United States. I suggest you contact an experienced immigration professional in your area to discuss this before you file any immigration petition.
They can get married, but it appears that he would have to process his green card from overseas depending on some specific circumstances.You will need to meet with an attorney with details. They lawyer may ask questions like whether someone has tried to file papers for your friends in the past, etc., to determine whether there are ways he can process his papers here without having to return home. Be sure to bring any and all immigration documents to the lawyer for review. If the person has to...
You may have various options to remaining in the U.S. You should visit a local immigration attorney with all the documents about your relationship, immigration, etc for review. The attorney may want to know details about the relationship before advice. It is sometimes difficult to analyze various options on this forum.