It seems pretty clear, to me at least, that the rich man was in what we know as hell while Abraham and Lazarus were in what we call the “Limbo of the Fathers” (please don’t say “we no longer believe in Limbo”- it’s the Limbo of Infants you are thinking of and even that remains a theological option, which in the end is all it ever was.)
I do not know if there can be any communication between those in heaven and those in hell, but in this case we seem to be looking at a case of what in the end are two “parts” of hell.
Regarding the natural goodness remaining in the rich man, I don’t think this should come as a surprise. Most everyone who goes to hell will have a significant degree of natural goodness in them at the moment of their deaths. Why should that goodness automatically disappear once they die? It might be that because of the experience of hell the damned go through a process of losing that goodness until at last they are much worse than they were when they died (that idea has been called Anti-Purgatory), but I’m not sure I buy it, and even if it were the case the rich man would have just recently died and so could have still held on to a natural love for his family even if he was later to lose it.
Not that I think the story likely literally happened, but at the same time I don’t think Jesus would introduce elements into a parable that would so confuse the listeners and later generations about nature of hell.