You've basically touched on the crux of the issue. Until about 11 years ago, the IGI baptisms were unknown. The evidence identifying Robert's wife as a Jeacockes (the spelling preferred by a Canadian branch, and by which I came to think of as the modern spelling) was not fully cited and the origins have not been recovered. My understanding is that Thomas Jaycocks referred to Robert Ashman as his brother-in-law, so everyone assumed that Thomas' wife was Ruth Ashman; then someone decided that Thomas had married an Armitage, so Robert's wife was an Armitage. Eventually, someone decided that it meant that Robert's wife was Thomas' sister. I am dealing with matters I last looked at 11 or 12 years ago and I haven't had time to dig out my notes. I have been trying the internet to see what evidence I could gather.
There were several notes on Robert's will, saying pretty much what you did, but I did find a fuller citation which cleared up some mysteries. This was at http://awt.ancestry.com/cgi-bin/sse.dll?db=gedoth&ti=0&f0=11674&f1=1468http://awt.ancestry.com/cgi-bin/sse.dll?db=gedoth&ti=0&f0=11674&f1=1468 and it quotes from a "Keeler-Wood Genealogy" pp. 62-3. It gives Robert's wife variously as Catern, Katrina and Katherine (an the opinion that she might be Dutch) and lists his children as:
Lydia Ashman m. ____ Swanson
Deborah Ashman m. Nathaniel Denton, Jr.
Mary Ashman m. Hope Carpenter
Martha m. _____ Cokse
Phebe Ashman m. Wait Smith
Hannah Ashman m. Thomas Flewelling
The double portion given to Thomas Flewelling, Jr. made me think that Thomas Jr. was an orphan in the care of his grandparents, possibly about 10 or 11 years old, but the 1673 date actually comes from genealogical notes on Hannah Smith's family (and were there indicated as conjecture. Nevertheless, they reinforced my ideas. Actually, you may be sort of quoting my theories back to me after having passed through several hands.
Your Passyunk reference is more detailed in, "Articles That Have Been Written About Carmans and Other Interesting Points", especially Helen Silvey's transcription of "Caleb Carmen: Whaler, Millwright and Miller", by H. Clifford Campin, Jr., 'The Cape May County Magazine of History', June, 1945, pp. 283-290 (http://home.att.net/~carman_family_history/articles/art25.htm. There, Thos. Hewellin becomes Thomas Filewellyn (a bit closer), and, as you pointed out, the Jacobs are Jacocks. The grant was 1 JAN 1667, but it is noted that Caleb Carmen disposed of his land in Hempstead in 1663. This coincides with Robert Ashman doing much the same about 1664, that is why it is assumed that they went to Passayunk about that time. Until now, it had been thought that Robert went directly to Jamaica, Long Island.
The significance of this is that it shows Thomas Flewelling, Sr. involved with Robert Ashman earlier than before expected, and presumably married to Hannah Ashman. Guesses on Thomas Flewelling, Jr.s birthdate had been based on a guess of Robert Ashman being b. ca. 1610, therefore Hannah b. ca. 1650 and her child ca. 1670; or variations. If thomas Flewelling was married to Hannah Ashman before 1667, then perhaps Thomas Jr.'s birthdate could be 1665/6. All, granted, theoretical, but it shakes some of the earlier assumptions. The other basis for guessing Thomas Flewelling, Jr.'s birtdate derive from his appearance in the 1698 census of Hempstead with three children. A guess was that he married at abou age 20, that each child was born each 1 1/2 years, making him about 25 in 1698, therefore b. ca. 1673. These have been repeated so many times that they have taken on the appearance of fact. The 1680 reference is also mentioned in the article and come from a document (transcribed in "New York Colonial Documents" Vol. XII, p. 648) and is dated 23APR1680, and lists Thomas Jacocks, Sr. and John Ashman, as well as the others. But, Ruth, Thomas' wife, is a widow by 1677, and John Ashman sold his 100 acres in 1670. Also Caleb and John Carmen are in Jamaica in 1673. The conclusion derived is that although the document is dated 1680, it's information was gathered about 1670. Their stay as a group in Philadelphia appears to have lasted at most only a few years.
As to whether John Ashman had children, I can only rely on the rumours which have come to me. There has been no evidence of the name, Ashman, in the areas where his children might have been, when they might have been.
Allen Dean Flewelling recently came across a transcript of a letter of Lars Persson Cock, Martha Ashman's husband. Notes with it indicate she married 15MAY1669 Lawrence Petersson Cock (b. 21MAR1646 New Sweden[i.e., in what became Pennsylvania] and that Martha was b. AUG1650 and d. after 1724. Another note attached to the transcript mentions Olaf 'Woola' Svensson, Lydia's husband. Allen found this on RootsWeb WorldConnect in a Rambo family tree.
I had not seen the variation, Ashmun, until you 'mentioned' it. For Robert's family, Ashman has been fairly consistent. Can't say that your variation hasn't appeared though.
SO, nothing proved, mysteries still abound, but perhaps some progress made. I would be interested in the Connecticut Ashmuns and the wills of Nathanial Denton and Hope Carpenter. This stuff probably won't help you come to any conclusions, but it might help you develop a theory to test. Drop a line on one of the Flewelling pages if you find something.
All the best,
Thomas A. Murray