What is a reasonable scholarly approach to breaking down short pieces of prose? Is it word choice? Is it structure? Is it themes, and how well they are used? Use of symbolism and its efficacy? Meter, rhythm, style, verbosity? Are these all on topic? Do you have a rigorous guide you use to approach short fiction when providing feedback to other writers?
You said scholarly, but I'll suggest something a bit less than academic (even if some knowledge of terminology will come in handy).
1. Read the whole story
This first step is applicable both to short stories and short novellas. Read the whole thing and simply enjoy the read.
2. It's about...
Identify the topic in few words (eg.: it's about a man trying overcome his grief) and the message (eg.: shy teenagers must develop the confidence to make friends and integrate themselves)
3. Go through the mechanics
Identify the main plot and the secondary plots (likely to exist in longer short stories and novellas).
If there is a secondary plot, does it help the main one progress or does it detract from it?
How are the different scenes organised? Do they follow one after the other, or is there alternance between scenes with one character or another?
How do the different scenes progress? Is there a steady pace or does it have slow and fast moments? In either case, is the speed appropriate?
Do note that some stories are meant to have a slow pace, so don't let a personal taste for fast paced stories tinge the slowness as bad per se.
3.2 Time and Space
Go back to the sequence of scenes and see when and where they are set. The evolution of time and place (or lack of it) has meaning.
Do the scenes follow a chronological time or do they jump to the past / future? What effect does it cause?
How is time perceived by the characters?
How is space described (private, social, ...)? Is there a difference in the way the narrator presents it and the way the characters perceive it?
Is there symbolism connected to the places the scenes are set in? Descriptions and even dialogue will hint at it.
Does the weather play a part in the setting?
Identify all the characters by importance (main, secondary, extras).
How are they characterised physically and psychologically? Is the characterisation done directly or indirectly?
What are their motivations? Are their ideas, atitudes and actions well aligned?
How to they develop (or not)?
Identify the type of narrator.
How does it impact the story?
If a 3rd person narator, does it have a strong voice or does it 'hide' under a character's POV?
3.5 Narration, Description and Dialogue
Are the paragraphs long, short, or a mix?
How important is the dialogue and how frequent?
How much description is there? Are some things / characters described while others aren't? How is it done?
This includes figures of speech, but also sentence size, adjective and adverb use, and pretty much anything that is used to convey a specific effect.
Think about what you enjoyed and what you disliked. What caused those feelings of like and dislike and why?
If you want to break down the text into sections, it depends on both the text and your object of analysis. Typically, a story will be broken down into arcs, but some short stories may instead be broken down into...
places: the story opens on a beach, progresses into a bedroom and ends back on the beach,
chronological time: the story starts with the morning, there's a change of pace into the afternoon and the conclusion happens in the evening,
psychological time: time moves slowly during a meeting, but then starts moving quickly once something is said,
... or other options! If you want to analyse a word fight, you may want to break it into moments of attack-defense and control. If the objective is symbols, then you identify all possible symbols and see how they relate to actions and moods (both of the character and of the text), then you break the text into the appropriate sections. One can even divide a single paragraph into moments if need be.
While analysing the text as a whole, you will come across both strong and weak points. If you decide to focus your analysis on a specific point (symbols, characters who do not act according to their supposed ideas / ideals, pacing, ...), you're likely to identify more weakness and strengths that are less noticeable at first 'read'.
My suggestion would be to list both the strengths and the weaknesses and present them as 'the setting feels off because ... ' rather than 'you failed to convey the right mood'. You may choose to always give a full analysis or focus on some aspects.
For me, a short story is a story. It follows the three-act structure, but it requires some inventiveness to compress that into a short space; in some cases to a line or two.
I don't care about metre or rhythm, a short story is not poetry. Nor do I care about "style", it is engaging or it isn't. Same for "theme".
Because it is compressed, unlike a novel (IMO) every line demands scrutiny and must advance its act. This puts more pressure on word choice and impactful action and at times evocative symbols. Where it is okay in a novel to describe something for a few paragraphs that isn't terribly important to the story, that kind of thing should be squeezed out of the short story. The same goes for idle conversation, in a novel you might like to hear two characters sparring or joking in a conversation that has no bearing on the plot, it is just "fan service" in a way, people like to read about some pair having some fun. Or character building, for those two. But you wouldn't write a whole chapter of that, and in a short story, the same conversation might be the equivalent of a chapter: There isn't room for it. In condensed form, proportionately from novel-length to short-story length, it might be one good line for each, but if that isn't enough to convey the sense of friendly sparring, it might best be left out altogether.
No, I don't have a rigorous approach to short fiction, just that it cannot be flabby, it should evolve logically and recognizably through the three acts (or four, or five if you like Shakespeare's style).
It should present a normal world, then a problem, that grows worse, but gets resolved, one way or another, with minor and major conflicts along the way to sustain reader interest. Just like any other story.
Unhappy or tragic endings are more acceptable in shorter forms; readers haven't spent many hours becoming emotionally invested in the characters.
I think all of what you just mentioned is on-topic for that type of critique. I also like to see if the author is looking for any specific critiques. For example, are they interested in hearing about everything but are specifically unsure about whether the story flows well? Is there a character whose dialogue they think might be stilted or strange? If I have a direction to look in, I'm much better at offering suggestions or commentary.