Do people use "facebook" and not something like "φασεμπούκ"? Thanks in advance
Yes because it's a site name. All sites (YouTube, Facebook, Twitter etc) use their original name. You can always write in with Greek characters though. Φέισμπουκ, γιουτιούμπ, τουίτερ etc. I, as a native, use φβ most of the times referring to Facebook, when chatting. But I think this is personal.