To build on what everyone else has said about the absence of the mother, etc., I frankly didn't think that deeply about it (but then again, I never took an English class in college). I just chalked up any of those "missing" elements or shortcuts to the fact that this was a short story and so Fitzgerald didn't take the time to flesh them out (and rather than do it in a halfhearted way, he just skipped them altogether). But as I read your thoughts, maybe I think that Fitzgerald purposely left those things out to leave them up to the reader to ponder. I would be fascinated to find his early drafts and notes to find out how things would have been filled in if this had been a novel.
Another dynamic that I thought about and would be curious to get your thoughts on (because if he did it on purpose, I think it was truly genius) is the sort of "bell curve" trajectory that Benjamin Button's life takes. In the normal course of events, a person is born with little knowledge and no experience, and through nature (instincts, genetic programming, etc.) and nurture (environment, parenting) he or she learns through experience and through study until he or she is a functioning member of society, and then his or her physical and mental abilities eventually begin to tail off (old age, forgetfulness, dementia, etc.), all of which is considered normal. In Button's case, his life also takes a similar bell curve, with a couple of distinct differences: He is born as a smart man but demonstrates a sort of "helplessness" in the role he's placed in because he doesn't have the instincts of an infant or any tendencies toward typical infant or childlike behavior, then he grows and transitions to a level in which his inner feelings, outward appearance and behavior fit with expectations for someone who looks like he does, and then as he gets older, he retreats into the mind of a child. It's as if the same sort of bell curve exists, and Fitzgerald has drawn parallels between Button's life and a normal life.