Why does Dickinson set the poem "Dying" in an ordinary bedroom, which isn't even described? What might be her purpose in choosing this setting as opposed to somewhere outside or in a public place? Is the choice of setting significant in any way?

I would think that setting the scene of someone dying is appropriate to be in a bedroom where a sick person would most likely be (if not in a hospital) such as someone in palliative care for example and the bedroom stillness is commented on broken only by a fly and whose buzz perhaps accentuated the otherwise stillness. If the person's death was outside or in a public place it would more likely be the result of a, say an accident, or say an assassination such as that of Martin Luther King so since it is inside in a bedroom it indicates the dying is just a normal death from illness and/or old age.