I just read the first two chapters of James Joyce and the Common Reader by William Powell Jones.
Basically Jones is interested in the linguistic/stylistic shifts between Joyces' various projects. To put it another way, he is very interested in how Joyce evolves as a writer, going from the realist Dubliners to the experimental Finnegan's Wake.
In his chapter on Dubliners he makes the following important points:
1) Jones believes that Joyce wrote Dublin as a microcosm of the universal experience. (basically what Hellsapoppin states). The specific scenes of Dubliners represents the universal experience, even as Joyce also seems very interested in capturing his youth and the specifics/particulars of Irish life back then as accurately as possible
2) Jones insists that the order of the stories in Dubliners is extremely important; they are organized loosely by broad themes/content. The first three stories for example ("The Sisters," "An Encounter," and "Araby") are about childhood and having an experience that forces you to grow up or realize just how much of a child you truly are depending on how you want to read it (since I think you can read the narrator of these stories in either way). They go through different stages of life from being a child to adult maturity as Hellsapoppin points out.
3) The order of the tales reveal that many of the stories are thematic counterparts to each other, which is why the ordering matters. For example, "Two Gallants" has a playboy take advantage of a servent girl to get her to steal something financially from her employer, while "The Boarding House" has a mother and daughter who basically setup a man to take advantage of him economically. They are parallels and opposites of each other. "Eveline" and "After the Race" are paired, one is about a girl who almost goes off to Buenos Aires with her lover, only to hide behind her responsibility, the other is about a boy who goes off on adventure with rich Continentals only to spend his father's money irresponsibly.
4) Jones notes that most of Joyce's stories in Dubliners are slice-of-life tales that center around the character's epiphany. He compares that to religious epiphany in the New Testament, but never really discusses it beyond that. The purpose of bringing it up is more for analogy.
5) Jones also sides with those who see these stories as centrally about paralysis with the exception of The Dead, which he sees a warm portrait of Mature Love.
6) Jones talks a lot about how Joyce thought of Dubliners as a history of Irish "moral life", but again never specifically ties this to Catholicism.
Jones is virtually silent in the first two chapters of his book about Joyces' Catholocism. The idea for "epiphany" that is so important to Joyces' structure with each tale might have originated in religious texts, but it is divorced from any real religious connotations. I suppose you could read Jones's comments on "The Dead" as a kind of symbolic ressurection from all the sins and decay of the other stories (You quote Jones as saying, '' 'The Dead' {reveals} a new theme, that of mature love, sympathetically portrayed. The geniality and warmth of the story, in spite of its title, seem to have been intended by the author to lift the collection [that is the earlier set of stories] from its 'special odour of corruption' ... '')
I'm assuming most of the Catholic reading comes from Gifford, however. I'm sure there is plenty of scholarship addressing how Joyces's Catholicism influenced his work. However, just because a theme is universal doesn't make it automatically a Catholic theme; all or most writers write universal themes to a certain degree. Simply because the etymology of Catholic once meant "Universal" is confusing etymology with current definition (which refers to a specific religion with specific practices). Also, saying that Joyce was interested in certain themes because of his Catholicism (probably true enough and a point I can agree with) is not the same as saying he was specifically attempting to write Catholic themes.
I also hope I am not coming off as being anti-Catholic, especially as a non-Catholic (we unfortunately live in a world where people can be insensitive to Christians). So I want to explain why I am pushing this a little more. I guess what is bothering me still is I want MORE out of the Catholic reading. I want to understand better how you [Hellsapoppin] see the Catholic motifs and symbolism adding up to and signifying a particular/specific meaning. It's all fine and dandy to point out that this story is doing this and that with the trinity, but it never tells me what you think Joyce is driving at by using that symbolism. That's what I really want to know.
Not to mention it seems to me that at times Joyce is extremely critical of both the church and his fellow Catholics as well as Protestants (and perhaps Jews too).
*** Main Implications/Lessons from Jones: I wish I had read Dubliners in the correct order seeing how important the stories ordering happens to be!!!

