Character Analysis in Dubliners

We quickly learn that Eliza is the sister of the now-departed Father Flynn, whose first name we also soon learn is James. It's clear that she had been taking care of Father Flynn in his old age and illness—likely after his first stroke.

This detail shows us that Father Flynn and his sister Eliza were born in Irishtown, a lower-class district in Dublin south of the river Liffey. This makes his acceptance into the Irish College of Rome more impressive considering the circumstances of his upbringing.

This question expresses concern over whether or not Father Flynn received his last rites, known as Extreme Unction, before his death. The question itself also indicates the possibility of doubt, which adds to the mystery surrounding him in this story. If the rite's were not administered, that would have meant that Father Flynn had committed a grievous transgression against the faith.

Joyce's choice of "truculent" here to describe the dead priest's face has interesting connotations. The word itself means "aggressively defiant," which gives the impression that the corpse has a rather mean-spirited or combative look on its face. Usually the dead are described as lying peacefully. Perhaps Joyce is signaling that Father Flynn died with some unresolved issue or problem.

A Rosicrucian is a member of a fraternity of religious mystics which traces its origins to ancient Egypt by way of the likely fictitious 15th-century German monk Father Christian Rosenkreutz. The 19th century had a revival of interest in mysticism and occult activity when many considered conventional church wisdom unsatisfactory.
The uncle is making a humorous and likely derisive comment on the boy's interests in the mysteries of religion. The uncle likely considers the boy's association with Father Flynn not very healthy for a young boy, preferring him to associate with other youths outside and to "take exercise."