Her second great-grandfather resided in Orum, Randers Amt (map below) in Denmark in the 1880 census (I think!), but may have been born in Viborg Amt. When he migrated through Liverpool to Philadelphia, he left his family in Denmark and they migrated a year later. Dorothy wondered if the family came out of Copenhagen or another European port.
I don't have any known Danish ancestry, but my grandsons do! I have not done any research in Danish records, but have done some in Norwegian records online, on microfilm and in books with some success. So I'm relatively uneducated about the topic. But I know that there are church records on microfilm, and census records, and other records in repositories or online.
I went looking for Denmark genealogy resources on the Internet, and found a treasure trove of How-To articles and Helps for education, and specific information for the two Danish Amts in the FamilySearch Catalog.
I thought I would list some of my finds here, in hopes that my readers might be able to add more information or contacts to help Dorothy.
UPDATED 3 p.m.: I found some great videos on FamilySearch about Danish church records, and that some are digitized and online. Here are links:
UPDATED 8 AM Friday: I also corrected the Norway reference in the first paragraph...how did that happen? My fingers don't always do what my brain thinks they should do! Thanks, Howland!