Thanks for the reminder to enter verifiable sources, rather than simply link back to the Family Search profile. I am so new to WikiTree, I had to go back and find how the system wanted sources linked, so hopefully what I have input helps out. Hopefully the link to Katina's profile in the original post will show the updates.
One cool thing is recognizing that her mother was listed as Bessie in 1940, likely an Americanized version of her Greek name, and Bessie was a widow at that point, so that prompted me to search for George's (father) death record between 1930 and 1940, which I was able to find, and it listed his parents' names. That was exciting, at least having names of another generation back, even if I haven't yet figured out where in Greece they lived.
I've begun looking at Ellis Island records, and came up with a possible match, with a spelling difference for George (Georges Margetis) for the year that I'm expecting, but no record of any of his family coming in through the same port several years later. I'm going by census records that he came first in 1910, and the family followed in 1915. I have no information, whether documented anywhere or even in family oral tradition, that gives me an idea which port of entry they used. Ellis Island was just a starting point, to start somewhere.
So, that's what I have so far. Thanks for your willingness to help!