Jan, I looked at the passport applications again on Ancestry.com. There were two applications, one in 1901 and one in 1913. In 1901 he gives his full name as Wilhelm Heinrich Hermann Hollmann. It says that he was born in Aurt Diepholtz Aschen county Hannover Germany on 9 May 1865 and that he immigrated on 9 May 1883 sailing from Bremerhaven. On this application he is also taking his son, William Hollman who was born 9 Oct 1889. He lists the ship that he sailed on to the US as the Fulda (the spelling of the ship's name was not completely legible.
Now on the 1913 passport application he gives his full name as Wilhelm Herman Heinrich Hollmann and has the same birth date of 9 May 1865 and born in Diepholtz, Hannover. He says that he came to this country on the ship Fulda on 9 May 1882 sailing from Bremen. Both application have his naturalization date as 26 April 1899 in Minden, Nebraska.
So, we have two different years for immigration, but it is just one year difference which he might have just gotten the year wrong on one of them. As for the birth month of his son, he might have gotten the month mixed up with the birth month of his daughter Hazel. You have that she was born in the month of October and the son William born in the month of September.
There are several postings on Ancestry that give a name to the fourth child as Frank and it just says that he died "young".
I forgot to look at the census records again. The 1900, 1910 and 1920 censuses give the immigration year.