This distinction between "his" and "her" ïs characteristic of English, but in Romance languages the standard 3rd person possessive determiner/adjective doesn't differentiate between them. In Romance languages, on the other hand, the number and gender of the thing "possessed" is indicated, a distinction not made in English. All these mean "his/her newspaper":
son journal (fr)
il suo giornale (it)
su periôdico (sp)
Most of the time we use possessives we aren't contrasting possession by one person with another, so this question doesn't really arise, but when they do want to specify in Romance languages, they often have alternatives available, just as we do when we want to specify between two "possessors" of the same sex.
I can't really answer your second question, but I've read that the third person "său" forms are more formal, and reading Wikipedia articles, my impression is that the use of "lui, ei, lor" is (much?) more common.