Well, the easiest way would be to administer 20 consecutive fill-in-the-blanks to get the names. Use variable names like name01, name02, ... name20. Then ask the 20 sets of follow-up questions and simply include the name in the question wording like "Tell us how <name01> makes you feel..."

If having the multiple questions on the same page is necessary, you can get into creating the item as an HTML form (see Creating Custom Items in the users guide).

There are a lot of options for doing to stuff like this, so feel free to let me know if you run into any trouble. The first method described above though would be super easy if you can live with the follow up questions being asked one at a time.