A: If your case was administratively closed, then it would no longer be possible for you to keep the original appointment. You should send a letter to USCIS requesting that your case be reopened, and an interview scheduled. Check the letter you received administratively closing the case and send your letter to the address listed there. You might have to follow up with an Infopass appointment, which you can request by calling the USCIS customer service number at 1 (800) 375-5283.
A: No, writing a letter of invitation would not make you financially responsible for this individual, there are certain application forms (Form I-134 or I-864) that deal specifically with financial responsibility. In addition, if your letter makes no mention of financial support, there is no basis for a consular officer to find that that you would be responsible. Your friend would have to establish to the satisfaction of the consular officer that she has sufficient income to support herself during her visit to the United States and sufficient ties to her home country to overcome the presumption of immigrant intent.
A: In order for an adoption to be binding on immigration authorities, the adoption must occur while the child is still under the age of 16 (along with some other requirements), therefore, if her son is already 23 years old any adoption would not be recognized for immigration purposes. But if you married his mother while he was under the age of 18, he would be considered your stepchild and you could file a petition for him as his stepparent. However, as he is already over the age of 21, he would no longer be classified as an immediate relative, which means you would have to wait for a visa to become available to him.