My Danish side of the family comes from mostly Shelby county Iowa, and lived all over in Denmark, but at least one family is from Hunderup, Ribe which I saw on your tree.
The Danish Archives site is in Danish, but is actually relatively simple once you know a few words of genealogy in Danish. The records that are on there are actual scanned images of the original record books. It's possible that the people at Family Search have not transcribed all the relevant records and/or they were illegible (which is sometimes the case).
I can't really help with the accents or additional letters. Those are still beyond me. When I was down in Harlan the last time digging up info, I didn't realize that the Danish History Museum was so close. I don't know when I'll be back, but I think that most of my family actually emigrated prior to the immigrant records being kept.
I love searching the archives, so if you want I can do a little digging and see if I can turn up any additional info. Let me know.