In all three examples, 'of' is performing a role indicating possession, though not in the way that of usually acts as a possessive/genitive indicator. This may be a source of confusion. In fact, they are indicating attributes rather than possession, but because we might say people or things have attributes, the matter can be confused.
So, here's a brief digression if you're interested in the normal sort of "possession". It's not really important to this answer - feel free to skip ahead. You would normally form possessives in English using a possessive pronoun (your, her, his) or the "Saxon genitive" -'s: Philip's, Barbara's, the Government's, and so on. In some cases, usually for poetic reasons or for the sake of euphony, you can use of - "the home of my aunt", "the banks of the Thames". (There are cases where the 'of' form is the only one available because there's no possessive version of, for example, some indicative pronouns).
However, in this case the thing before the 'of' is what possesses the thing after it - but as a characteristic, rather than an actual possession.
So, to answer your question, In phrases like these, "A of B" indicates that B is some sort of characteristic or attribute of A. "A man of about my age" is a man whose age is about the same as mine. A "country of wide lawns" is a country (which may mean an area of land that is not actually "a country" in the usual sense) which is characterised by wide lawns. A "gentleman of that name" is a gentleman whose name is whatever the indicative "that name" refers to.
Similarly, a "shape of ten sides" is a shape characterised by having ten sides (a decagon). A "bill of £42.50" is a bill stating that the amount owed is £42.50. Even the famous "writ of habeas corpus" is a legal writ to demand that whoever is holding a person produce them in court (habeas corpus meaning approximately "that you have the body", which is to say the person). A "woman of uncommon intellect" is an unusually smart woman, and a "man of habitual lateness" is a man who has difficulty turning up for anything on time.