We will be flying into Dubrovnik from Frankfort. Will we have to go through customs and passport control if we do it in Frankfort (coming from US)?
I guess you mean Frankfurt, Germany rather than Frankfort, Kentucky. As Croatia is not a member of the Schengen Area yet, yes you will have to go through customs and passport control. It's usually no big deal, and even if you are the last one off the plane and at the back of the queue, it is unlikely to take longer than 15 minutes.