In addition to many great Romantic texts, the mid-nineteenth century saw the rise of literary criticism. What do these writers think makes a good story? How do stories such as "The Fall of the House of Usher," "The Black Cat,” and "Bartleby, the Scrivener" hold up under their own criteria, and what makes literary criticism like this significant to the study of literature as an academic discipline?

Edgar Allan Poe himself was probably the most significant US literary critic of the early- to mid-nineteenth century. So your question partly involves whether in his own writings he followed his critical principles. In his review of Nathaniel Hawthorne's Twice-Told Tales, Poe speaks of "unity of effect or impression" as one of the most important features of the ideal short story. For the reader to reach a state of "exaltation of the soul" a prose narrative must be brief and create a single effect. If brevity is too extreme, the work degenerates into mere "epigrammatism," but to Poe, extreme length is an even more "unpardonable" error.
Like Hawthorne's works he is praising, the two tales by Poe to which you allude do follow these principles. If anything, however, one sometimes feels that in focusing on brevity as a virtue, Poe's stories are sometimes too brief or are not developed enough. In "The Black Cat" we learn little about the narrator beyond his own account of himself, which could be, and probably is, distorted by his psychotic mental state. However, one could say that this brevity and single-mindedness simply enhance the atmosphere of horror. Melville's "Bartleby" is quite different. The story is multi-layered and extended, but even so it follows Poe's prescriptions in giving a single effect, building up to a focused climax in the deterioration of the central character.
British literary critics at the height of the Romantic era such as Shelley, Coleridge, Hazlitt, and Charles Lamb stressed, as Poe would decades later, the need for the reader to be placed into a heightened or exalted mental state. This corresponds to the definition of "the sublime" which had been central to European literary criticism since the late eighteenth century. In his review of Hawthorne, Poe analogizes the poem with the short story as genres. Shelley's greatest poems, like Poe's tales, present a single effect that transports the reader into a meta-world beyond ordinary earthly life. This is the essence of the nineteenth-century literary aesthetic.