I have tons of genealogical information on the Streeter line dating back to Goudherst, Kent, England - and Stephen Streeter's arrival to Gloucester, Mass., around 1639-40 (married Ursula 1652). Let me know if you have any questions on Benjamin Streeter and descendants. I think I can answer a lot. I also have a subscription to Ancestry.com, not that this has been a big help.
My Dad has been looking to see if there was a connection to Cap Streeter for many years. His Father mentioned Cap as an uncle to my Great-Grandfather, Charles, but my father does not know how they all relate. However, if Cap was an "Uncle" to Charles, Cap (b. 1837) would have been around the same age as Charles' Father, Lorenzo (b. 1838). So, to reiterate what you wrote: Elias/Lovisa (first wife) had Thomas (b. 1804), half brother to Lorenzo (b. 1838) (my great-great-grandfather, son of Elias/Eva [second wife]) Around 1805, William Andrew (WA) Streeter was possibly born to John and Julia (not sure if they were Streeters) - William Andrew was then adopted by Elias - however, we don't have definitive proof that his biological parents were in the Streeter family. Thomas (b. 1804) and WA (b. 1805) had jointly paid taxes in 1850. Do we have any other proof to point to WA not being the son of Elias, but John and Julia, other than those names being listed on the death certificate? Yes. In my genealogy book from 1896, WA is not listed as a son to either Lovisa or Eva. I'm unable to find any "William Andrews," as such.
I do wish I could find out the connection between Elias and William Andrew. My Dad would be thrilled either way. And since we do not know the lineage of John and Julia - who knows - Cap Streeter may not have been a Streeter at all - but that would be a long shot.
Thanks so much for your help. Please let me know if you have any questions - perhaps we can solve this. :-)