The Problem with Italians

And probably with a lot of other ethnicities, too.

Imagine a family consisting of a widowed father and his 5 children: Domenico, Vincenzo, Rosa, Angelica, Maria, and Giovanna. The census taker can't spell their last name, D'Ingeo; the only time you've found them on a census, it's spelled Dengao.

Vincenzo, in that unusual trick of Americanizing the name Vincenzo, goes by James, or Jimmy.
Rosa can be Rosa or Rosie.
Angelica might go by the Italian nickname my grandfather pronounces "Yaneen" or "Aneen;" she may also take on an American nickname the way her siblings did, but I have yet to discover it.
Maria, of course, is sometimes Mary.
Giovanna is Giovannine in Italian; she's Jenny in English.
Their father, of course, could be Dominic, Dominick, Domenico, Dominico, Dom, etc.

They immigrated in 1909 (Jimmy), 1911 (Rosa) and 1917 (the rest of the family). Some portion of them should show up on the 1910 Census (I've found no one), the 1915 NYS Census (no one), the 1920 Census (I may have found Maria, grown-up and married to my great-grandfather), and the 1925 NYS Census (no one). They finally start showing up in 1930, grown-up and married, where I've found Jimmy with his family, Jenny with her family, Maria with her family, and Domenico with Jenny's family.

I simply can't figure out how to search for them more thoroughly. Usually I'd resort to exact-search functions and searching for the relationships between family members, but between the misspellings and the nicknames, the permutations are absolutely endless and I haven't had any luck.

Who has a tip who can get me back on track with these troublesome Italians?