What is Facebook's market? Social media or advertising? If the main "clients/users" are the typical average Joe, and the service is free, does it count as a monopoly (At least in terms of anti-trust laws)?

That it doesn't own? In the US you could probably could argue:

LinkedIn, Twitter, Snapchat, Youtube, Pinterest, Reddit, Tumblr. I be they would also argue Google, which would be true from an advertising perspective.