You are the family I'm looking for and you were right about being unsure about your facts.Your grandfather Rudolph Friedrich was born in Hamburg, Germany. Your great grandfather Wilhelm August Neumann was born in Gratz, Posen, Prussia which was and is again Poland. How do I know this, I have Wilhelm August's Birth Certificate and various other ones in their original language (German). I also have a document (in two languages German & Wendish) which we think was a permit to travel from Gratz to Hamburg through Wendish Territory. Rudolph Friedrich married Margaret Hughes 1906, so she was your grandmother.
If your brother is the genealogist of your family, I would dearly love to get in touch with him as I can supply him, for all of you, copies of original German Birth, Death, Marriage & Christening Certificates as well as their translations.
My grandfather Wilhelm Emil was your grandfather's older brother.They also had a sister Gertrude Emilie.
Looking forward very much to hearing from you and your brother.