We know that Jacob C. Finstad arrived in the US in 1871. This is listed in three census records. The family I am following definately arrived in the US in 1871. Here they are leaving Finstad for America on 27 Mar 1871 (right column, top entry):
Farmer Casper Hansen Finstad and his wife Maren Caroline Gulbrandsdatter and son, also his children by his first wife, Bertha Christine Hansdatter.
1) Hans Jacob, born 18 Dec 1858 (should read 15 Nov 1858)
2) Josephine Marie, born 12 Aug 1860
3) Ole Christian, born 18 OCt 1861
from second marriage
4) Johan Nicolay, born 2 Jun 1869
Kildeinformasjon: Akershus fylke, Nes i Nes, Ministerialbok nr. I 8 (1859-1874), Utflyttede 1871, side 511.
Permanent sidelenke: http://www.arkivverket.no/URN:kb_read?idx_kildeid=535&id...
I am now 99% sure that this is the correct family.