What is the role and character of a FATHER (biological and not) in a stories of Dubliners?
Focus on such stories as "An Encounter," "After the Race," "Eveline," "Counterparts," and "The Boarding House."
This is a very interesting question on a topic that I hadn't thought much about before. Eveline's father and Farringdon (Farrington?) in 'Counterparts' are perhaps the nastiest examples of fatherhood in the collection. Eveline is cowed and dominated by her very demanding father and yet she not only clings to a couple of pleasant childhood memories of him but ultimately cannot leave him because he might really need her. Farringdon, the alcoholic father who spends a wasteful day avoiding work and going in and out of pubs, finally arrives home at night to take out all the frustrations of his day on his innocent and pitiful child. It is a terrible scene, possibly the most heart-wrenching in the collection.
No father features directly in the other stories you mention but certainly in two of them the influence of a father is there. Jimmy Doyle, through his father's wealth, is able to participate in a lifestyle that dazzles and flatters him but he is really out of his depth in one way and in another would do far better to steer clear of this company of wastrels for his own good. His father rather misguidedly indulges Jimmy and, although his fatherhood is a far cry from the two mentioned above it is still damaging to the son.
In 'The Boarding House' the landlady's father (can't remember her name off hand) was a butcher and it is strongly suggested that she has inherited a very pragmatic approach to life from him: she cuts through things briskly, efficiently and singlemindedly. The lodger that she decides will marry her daughter is like a lamb to the slaughter once she has decided his fate. I used to consider this a humorous story but re-reading it more recently I found myself much more in sympathy with the lodger than before as the sense of his victimhood and being carved and parcelled up like a purchase of meat came across more strongly to me.
I can't think of too much to say regarding fatherhood in the other story you mention, 'An Encounter' but it did strike me that the absence of a father in the early stories of the collection could be significant. The boy in 'The Sisters', 'An Encounter' and 'Araby' is something of a loner, a child who lives in the world of his imagination, perhaps more so than a boy with a 'normal' father might, where the father would perhaps provide a role model and offer alternative pastimes or activities. Not that there is anything wrong with living in the imagination, but just that things might have been different with a father figure present.
Hope this is of some little assistance and thanks for the question: it certainly made me think about 'Dubliners' from a fresh perspective and, since it's not far off 50 years since I first read the collection, that's saying something!