Computer space

does face book and all the interaction between users, like my daughter and her friends, take up space in my computer

No, not unless they are saving information via documents, or images.

Everything on those kinds of sites are on their own server, and most likely the only thing you'll find on your computer are temporary internet files and cookies, which use up next to no space.

Not really. The only thing from any Internet browsing that takes up space is cookies and the temp files. Also, if they saved any pics etc onto your computer. But that is something they would have to do themselves. The computer doesn't do it automatically.