I looked through the Danish Census records but that area does not have census records beyond 1845. There are many Carstensens in Tønder, Løgumkloster. I did find a Niels on the Emigrant database. It says, "USA Carstensen, Niels Andersen* 1858 Accused (in 1893) of not showing for military service and of leaving the country without the required permit. Reserve soldier."
If this is your Niels, his father's first name was probably Anders. This Niels went to the USA, was born in 1858 and left before 1893. I didn't find any information on Dora or Doris on that site. If I find anything else, I'll post it.