According to the author herself,
adamant pretty much has it right [that is, in saying that "the existence of three different pronouns need not have much to do with physical sex"]. Genitals don't define gender for Hwaeans. Also, this particular construction of gender is common/dominant in Hwae and a few other systems.
(source)
Nearly all of the questions given in the question overall are answered by this statement. My comments, not those of the author, in italics:
What's up with sex and pronouns in Ann Leckie's Provenance?
Nothing is up with sex and pronouns; but pronouns are added to describe a third gender.
I'm getting the impression that maybe there are three sexes...?
No, but there are three genders (sets of social roles).
Are the pronouns like "e" and "eir" for a neuter gender?
They are for a third gender, yes; but not precisely for a neuter gender (since the other genders, having nothing to do with sex per se, aren't precisely "masculine" or "feminine".
What is a "neman" or a "nuncle?"
Leckie doesn't directly answer this question in her comment. I'm trying to clarify with her.
Do people choose their sexual identity when they become adults?
Not their sexual identity. Leckie doesn't explain whether this is a choice, or when it's made, if it is a choice.
is this a physical (surgical?) choice, or just some kind of choice of what clothes to wear?
Since, as she says, this is nothing to do with genitals, it is certainly not a surgical choice.
Is the linguistic stuff just something about the culture of the protagonist, or is it language being used to describe some set of biological or social facts that are present in other cultures as well?
"Genitals don't define gender for Hwaeans.... this particular construction of gender is common/dominant in Hwae and a few other systems." That is, the linguistic constructs describe a system of cultural and social facts not specific to the protagonist or their culture.
