There are 3 speakers: the mariner, the marginal gloss, and Coleridge. What are each of their purposes and understanding in the poem?My professor...

Some of what your professor says I agree with, some I do not.  Let's look at the 3 speakers she says that there are in the poem.

First is the mariner, who is fated to tell his story for the rest of his days to the people who "need it."  The mariner, in my opinion, does completely understand why he has this purpose and why he is doomed to roam the Earth repeating his story.  He knows the wrong he has committed and he knows he is paying for it through his re-telling of his story.

Second, there is the narrator, not necessarily the poet himself.  We cannot assume that Coleridge is the narrator.  I separate the narrator and author completely and see the narrator's purpose to present the story objectively, as an outsider looking in.  If the poem were narrated by the mariner or the guest, for example, we would get two completely different views and ones that are subjective.

Lastly is the marginal gloss, which is full of definitions and explanations and notes. It disrupts the flow of the poem in ways and could definitely represent causality, as your professor says. 

I do believe there is definitely a theme not only of respecting nature and God's creatures, but one of repentance and salvation, in addition to one that involves the role that fate/chance plays.  The mariner's fate is that he must re-tell his story over and over again as part of his pentance and to help relay the moral of his story.