Thanks for your reply. I don't have any birth, death, or marriage certificates - yet.
In 1920, Charles and his second wife, Ida, are in Pillager, Cass County.
Hannah, Peter Carl, and Viola are his children with his first wife, Anna Nyman.
Ingrid (my grandmother) and Victor are his and Ida's children.
According to this census, he was born about 1868, immigrated in 1877, and was naturalized in 1904.
These children all match the information I got from my father, and I know that his grandparents lived in Pillager, so this census record (and the 1940 one) is the most solid one I've got so far.
I haven't found the family in 1930, but I know that sometime around then they were in or around Cloquet, MN, where my grandmother attended high school.
I'm less convinced that the 1910 census record you mention is my Charles and Ida; Dad has never mentioned an Aunt Annie (though I realize she could have died before he was born). Also, Charles's immigration date here doesn't match the information on the 1920 census. Carl P in that census could be Peter Carl in the 1920 one, and Mabel V might be Viola, but I'm not sure how much that's stretching things.
I just found a 1900 census record, though, where Charles and his wife Anna C are living in Rail Prairie, Morrison county, next to Peter and Anna Nyman, likely her parents. This record gives Charles's birth as March 1869, which fits with the 1920 and 1940 records (the two I'm most confident in so far), and their marriage as 1897. It sounds like a trip to the county recorder in Little Falls is probably my best next step.