Just adding what info I have been told:
I spoke with my grandfather, whose father was Julius William Picaman. My grandfather said that he had been told by Julius (and throughout his life) that Susan was full-blood Cherokee. Additionally, my grandfather spoke with John William Picaman when John was old and my grandfather was young. Grandfather said that John told him "He and his brother, Jake, stowed away on a freighter to America from Germany when John was 17 and Jake was 15. They were discovered mid-voyage and had to work as cabin boys until they arrived in America. Then they 'jumped ship' and changed their surnames, John to Picaman and Jake to Pickerman." Grandad did say he could have some of the details wrong--he remembers a freighter, but it may have been a different type of ship. Also, they may not have been 15 and 17, but the story doesn't seem to work unless they were close to that age. John also never mentioned another brother to my grandfather as far as he remembers.
My grandfather also met "Uncle Jake", as he called Jake, in Paducah, KY when he was a boy. Granddad said John never talked about Germany or his parents, but during WWII, John read any news articles he could find about Germany. Granddad remembers him as having an eidetic (or possibly just extremely good) memory, as John could and did quote entire news articles if he was prompted by a comment someone would make (e.g. mentioning a news event John had read about). Also, John had one real eye and one glass eye. As granddad remembers, he had been working with metal or slag, possibly making horseshoes, and a piece of slag flew up and put his eye out.
John also built two log cabins near Claremore, one he lived in and the other my granddad lived in when he was a boy. Reportedly, both are still standing and still have residents. Grandad also said John built the first cannery in Oklahoma, along with a sorghum press, which grandfather had used when he was a boy. John was also a board member for Harmony Star school, 1/2 mile or so from granddad's house. Susan is buried on the south side of a hill some distance southwest of the school, near the first log cabin John built.
All this is as told to me by grandfather. Some details may be inaccurate, but hopefully this sheds more light on the life of John William Picaman.
I would be interested in seeing anyone else's information, particularly the source for three brothers immigrating specifically from Hamburg, Germany. This information is new to me.