This answer mainly deals with your Q1:
What does she mean by 'Firmament to Fin?' My guess is that fin is an alternate spelling of fen, so this is along the lines of saying 'as foreign as heaven is to earth.'
I smile when you suggest that I delay "to publish" -- that being foreign to my thought, as Firmament to Fin.
The key is that to delay publishing is foreign to her, as foreign as firmament is to fin.
firmament noun (literary)
The heavens or sky.
The firmament is a reference to the sky, and fin is a metonymic reference to fish. Just as the sky is a foreign environment to fish, an intentional delay in publishing is foreign to Dickinson.
Alternatively, it has been suggested that Dickinson may not have been interested in publishing, simply seeking an established poet's opinion of her work. This would be consistent with the next 2 lines (incidentally, your Q2 and Q3), which refer to her lack of fame and inability to pursue it. In that case, it would be publishing that was foreign to her. Either way, "as firmament to fin" refers to something foreign, or somewhat akin to the idiom "a fish out of water".
As for your Q2, in the phrase
the longest day would pass me on the chase
the imagery is that of a fox-hunt as you suggest, where she hunts fame. Her assertion was that she couldn't escape fame if she was already famous, but if she was hunting for fame, she would fail. That is, she thought she wouldn't be able to capture fame, even given the "longest day", or the maximum time allocated to her attempt (presumably, to the disappointment of her dog, who would have participated in the hunt).
Regarding the barefoot ranking of your Q3:
More importantly, what if she not only meant what she wrote, but what if she was right? Is there a sense in which a “barefoot” ranking is, actually, better than fame and a public career in the art of poetry?
- David Graham, "My Barefoot Rank"
Of the 3 questions, I'm least sure of my answer to your Q3. However, the comparison appears to be between a professional ranking conferred by scholars, and a 'barefoot' ranking among the general population.