Hi Zachary. I enjoy working on other researchers' brick walls from time to time, rather than just continuing to bang my head against my own brick walls. The problem of your William's parentage is intriguing and so I've continued to think about the various issues raised by the evidence you cited and the additional material I've found.
As your original posting also asked for comments on various theories, I've done some further poking about and will offer my opinions hoping that also may be of some use to you.
As you note, there are apparent errors and inconsistencies in many of the records pertaining to your William. Over the years I have found that there are going to be errors and inconsistencies in almost every genealogical question, but you have certainly had to deal with more than your share here.
My comments need be a bit long to try to tie up the strands of evidence, so I'll break things into separate postings, one for each of the three theories you posted about, as well as one more for my own conclusions and to sum up.
I'll start with Theory #2, Daniel & Harriet. The primary evidence to support them as your William's parents is that the 1850 and 1860 censuses for their household shows a male child of the same first and last name, William Flower, born about the right year in the right state (Michigan). In 1850 he is listed as William H. Flower, and in 1860 as Wm Flower. You gave the link for the 1860 above; you may already also have the 1850, but if not, it can be seen here:
https://www.familysearch.org/ark:/61903/3:1:S3HT-D1M3-GYJ?i=21&cc=1401638https://www.familysearch.org/ark:/61903/3:1:S3HT-D1M3-GYJ?i=21&cc=1401638
I can see why you found this set of possible parents so intriguing beyond the simple name/age/birth state match. Harriet was born in New York, one of the two different places (the other being Prussia/Germany) that some known records about your William suggest may be his parents' birthplace. In addition, the William associated with this couple disappears from Knox County Illinois census records after the 1860 census, which is consistent with your William's known arrival date in Iowa in the early 1860s. And, one of Harriet's parents was apparently born in Germany. If these were his parents, it would neatly tie up a lot of loose ends.
On closer examination after some further digging, however, I think it is quite clear that Daniel and Harriet aren't the parents of your William. I won't discuss all of the circumstantial evidence against it, but will concentrate on one key point. Specifically, I believe there is compelling evidence that the William who is shown in the 1850 and 1860 census with this couple died on April 13, 1865, and thus cannot be your William, who lived on to 1930 in Iowa.
In the same cemetery where Daniel and Harriet are buried, Hope Cemetery in Knox County Illinois, there is a military gravestone for a William H. Flower, with no dates, but inscribed simply "WILLIAM H. FLOWER / Co. E / 17 ILL INF." See
https://www.findagrave.com/memorial/15170440/william-h_-flower#view-photo=3734615.
The online database of Illinois civil war muster roll data at the website of the Illinois Secretary of State shows that a William H. Flowers from Galesburg enlisted in December 1863 as an Illinois soldier at age 18. It further states that he had been born in Kalamazoo, Michigan. At some unstated point in time, he was transferred from the 17th Illinois Infantry to the 8th Illinois Infantry, and died on 13 April 1865. See:
https://www.ilsos.gov/isaveterans/civilMusterSearch.do?key=84170.
What even more clearly connects him to Daniel and Harriet, beyond the obvious circumstantial evidence of his age, name, and residence, is that this same soldier's service was the basis for a survivor's pension granted to the soldier's mother, Harriet Flower. See
https://www.familysearch.org/ark:/61903/3:1:33S7-9TBN-FJG?=1909&cc=1919699.
There is more evidence which further substantiates that the Harriet who received this pension is the Harriet who was Daniel Flower's wife, but hopefully I've provided enough here to set the question to rest. If not, let me know and I'll post the additional details.
I'll discuss Theories 1 and 3 in a further reply (but not today).
