Regarding social networks and your second question,
What properties/quantity people want to know about them
You can try it yourself, so anything you might want to know about the social network is a valid answer to your own question.
Ref How to use Facebook's Graph Search (and why you would even want to)
Some examples given in the article are
I searched for photos of my friends who visited Hawaii, because I just returned from a trip to Oahu and wanted to see their photos of the islands.
Friends who listen to Daft Punk and live in San Francisco
Friends of my friends who work at TechHive
You can also find some other real world applications from the websites of Palantir and Neo4j
I remember some years ago seeing a presentation from Palantir where they helped the police use graphs to model criminal networks. So they would add connections between people based on friendship, family relationships, financial transactions, phone calls, etc, as well as businesses used for money laundering, where they kept their bank accounts, etc. I think the police has done similar work on paper for many years, but with graphs and computers it can be made more efficient, scalable, and easily be queried.