Thank you for reaching out. All the names you mention of the DeMartini descendants are ones I am aware of. Your grandmother was first cousin to my grandmother's father (Edmond DeMartini by his father John Baptise and his brother Frank, your great-grandfather).
Were you at the family gathering in Half Moon Bay at the old DeMartini ranch back in 1997? I was in high school then, but I meet many of the family there, and if you were there we may have meet before. I was compiling family names and sent copies to those who asked. I went to high school with Rosemarie's granddaughter, Maria.
Before I forget, as I was reading you post, I noticed one difference... Paolo [Paul] (your grandmother's grandfather) married Catherine, not Mary. It MAY be possible that Maria (Italian form of Mary) was a praenomen or middle name but Catherine was the given name used in Census record and from info my grandmother had.
Speaking of Italian form of names... You mentioned Gio Batta may have a brother George. I had record that Giorgio Batta was the husband of Dina Amelia. Giorgio is George in Italian and commonly shorten to Gio, so Gio Batta may be the same person as the George you are thinking is a brother.
Grazie e ciao