In the 1900 census, James Barta is indexed as James Burta. The family's census image can be seen in South Omaha, 0111, Page 37 of 46, line 37. Katarina is the mother of 5, 5 living. Birthdates: James/Vaclav Oct 1863; Katarine Dec 1866; Mary Apr 1894; Rudolph Apr 1895; Alfred Mar 1897; Annie Jan 1898; Joseph Dec 1899. James and Katarina have been married 7 years. They are living next door to the James (Vaclav) Zadina family.
In the 1910 census, his name is listed as Vaclav, his wife as Kattarina. The family's census image can be seen in South Omaha, 0116, Page 2 of 21, line 89. Katarina is the mother of 8, 7 living. There are several discrepancies: married 24 years (not likely); birth year of Rudolph (1891); birth year of Alfred (1893). Children still at home are Rudolph 19, Alfred 17, Annie 13, Joe 10, Louis 5, and Emil 2. The deceased child is not in the 1900 census. I have a record from St. Mary's cemetery that has a child, John, age 1, who died Jan 21, 1904, and who is buried in the same plot as Vaclav and Katarina.
There are subtle pieces of data in the 1910 census, however. The family is living close to the Trouba family, and I know that the Troubas and the Zadinas are related somehow. The area where they were living, 39th & Madison, was home to many working in the packing plants. According to my Czech teacher, who is descended from the Zadinas, it was also known as "Little Čáslav" because so many of the residents there were from the Čáslav area in Bohemia. For this reason, I feel that the location for Katarina Zadina's birth was near Čáslav in Bohemia. It is very close to Kutna Hora, which is much closer to Prague than to Brno, which is in Moravia. I took a look at the records in actapublica for Čáslav but did not find either Vaclav or Katarina. The Čáslav books do not include the surrounding villages, so the next step would be to search those towns to see if either the Bartas or the Zadinas lived there instead of directly in Čáslav.
I have not followed through on the sibling information listed in the original post, although the 1900 Zadina family has children Louis, John, and Mary.