Thank you for all the advice and research.
Is there a reason why I am only finding death and immigration records on Ancestry.com, but never marriage or birth records? Am I looking in the wrong place, or am I looking wrong? Are New Jersey records not as widely available as, say, Ohio records? Aside from the census and possible arrival records for Klara and Josefa, I have not seen anything on them.
Josefa was married in 1919 in New Jersey and promptly moved to Ohio. From 1920 on, her life is well documented on Ancestry.com, in the local newspaper (available on Google News Archives) and family memories.
As to the link you provided explaining manifest markings, it's invaluable! I have always wondered about those and did not know where to turn for explanations. I even had a family (three people) crossed out from one of the manifests (a different family), and I still entered them into my data as having traveled that day, because I assumed (incorrectly) that the manifests were compiled upon arrival or while in transit (sort of how we fill out customs declarations on the plane before landing), and the presence of the names on the manifest could not be explained other then by the fact that they sailed on it. I guessed (wrong again) that the crossed out lines meant something like "done processing these" upon arrival.
And you are right, these marking do help figure out whether and when the person was naturalized. I had no idea what to make of them.