What you have written here are nice family stories, but it is not documentation. For instance, you write:
"Solomon came to America as a translator; he returned to Russia to bring his wife and two children, Mike and Myer here in 1901."
However, you also write that Myer was born in 1903 in Russia.
I believe that Mike, Myer & their Mom arrived in NY in 1906. Husband Solomon was not on the ship with them, so he apparently did not return to Russia to bring them here.
The names are:
Kuperschmidt, Golde, b. circa 1880
Kuperschmidt, Meier, b. circa 1905
Kuperschmidt, Meisch, b. circa 1903
Arriving May 22, 1906.
They are from "Bialyczerkow" which I believe may be
Bila Tserkva, Ukraine, which was in Kiev province prior to WW1.
They were bound for Golde's husband - S. Coopersmith in NY.
Manifest can be seen at:
Begin your research with what you know, and go back slowly generation by generation. Document everything with vital records, census, ship manifests, naturalization papers, etc. After doing all that you can begin to search for records from the town of origin.
For instance, the family is easily found on the 1925 NY State census, and Solomon Coopersmith's court of naturalization is filled in there.
Good luck in your search,