Louis Carson (real name: Lutje Jakobs Karst) was born 29 March 1847 in Westernieland, Gemeente-Eenrum, Groningen, the Netherlands, son of Jakob Rikkerts Karst and Antje Willems Schaver. Between 1870-1873 he married Tryntje/Trientje Veen:
Marriage in County of Ottawa, Michigan: VEEN, TRYNTJE; VERSEN (KERSEN), LUITJE; Volume 5, page 8.
Trientje Veen was born in Michigan (appears to be Ottawa County) and she died in Missaukee County before 1893.
They had 5 children: Antje, Cornelis, Jacob, Roelof (1881-1883) and Roelofke (your Rose).
On 15 February 1893 in Missaukee County, Lutje (Louis) married a second time to Pietje (Nellie) Jelles van der Kam (her last name is her maiden name). In the Netherlands she was married to Albert Stevens Dijkstra. Albert died in the Netherlands in 1887. Albert and Pietje had a son Jelle Alberts Dijkstra (in America the last name is spelled Dykstra) who married the sister of Roelofke (Rose), which was Antje/Anna. So Anna Carson married Jelle Dykstra while the father of Anna and the mother of Jelle married each other.
So, none of Rose's brothers or her sister were born in the Netherlands. And Lutje/Louis and his second wife Pietje/Nellie did not have children together. Pietje/Nellie died 21 August 1922 in Missaukee County. Lutje/Louis died 24 June 1920 in Missaukee County. They are both buried in the Richland Cemetery.