his 3rd class family?

That paper-signing scene doesn't seem to make any sense. I once thought they might be immigration documents but that couldn't be it -- Sturgis only purchased the father's ticket, not his identity. He had his own passport and identification so it's not like he would have to pass himself off as Mr. Euskadum (or whatever his name was). The wife and kids meanwhile would have had their own passports and documents and so didn't need the father to be with them or sign for them; Sturgis in any case could confirm that he had purchased the ticket if necessary.

But in that scene Mrs. E. seemed to be conferring some right on Sturgis, in her alleged capacity as head of the family (which she protests she's not -- that he is, which really makes no sense). What could she possibly be doing that would involve him? And for what purpose?

And aside from what the papers might be, where did they come from? Who prepared them, and why? Did White Star have a resident attorney aboard who could draw up legal documents on the spot? None of that is answered or makes sense.

I think this is just some weak and unexplained plot ploy to make Sturgis seem responsible for the family's welfare, so that he has additional reason to go below and get them off the ship when it sinks. But there is no real sense or logic to anything we see.

You also touched on a shameless aspect of this scene, mogul: the sight of this woman, needing help with these mysterious papers, simply getting up from her desk in Third Class and going upstairs to First to ask for Sturgis. Of course, in real life there was no possible way she would have been allowed into First under any circumstances.

This is part and parcel of the overall tone of the film, in which the classes more or less mingle unobtrusively and where the rich First Class passengers are mainly depicted as cheery, selfless and generous sorts, while Steerage passengers are seen as naïve, simple, salt-of-the-Earth halfwits who need to be told what to do by their betters...including even getting off a sinking ship. (Remember when Sturgis goes down to rescue his "other family" and he and Sandy are told that the crew can't make them understand about the danger and they're all so panicked and confused they refuse to move?) We've discussed this elsewhere before, but this is clearly the doing of screenwriter and producer Charles Brackett, a very conservative man who always depicted the rich (pardon me, the job creators) in a positive light and the poor as decent but hapless dolts who need a guiding hand to help them survive. He was never more deceitful and shameless in this respect than in fashioning his version of Titanic, a despicable insult to the truth about the classes aboard that doomed liner.