It's a layout flow convention. You'll have the logo on the upper-left area of your layout because the western eye starts to read from top left to bottom right. Then, one of the ways to add other important information (and it's NOT the only one) is to use the next available zone, which is top right.
Some sites use a RIGHT sidebar, and while it may feel unnatural for western users, it really doesn't have much issues either and people in large screens tends to adapt to it really fast. So, in the end, it will depend on your layout design decision.
As for where did this start, it really didn't start with Facebook, sites in mid 90s already used this approach (with or without avatar, I mean the positioning) because, like I said, it's the most natural zone in your layout to place that info.
Finally, about your first name / last name question, the most appropriate way should be to use a "Name to Display" field, so people can choose which name to display. This is important because some people don't like their names and some people has issues with privacy. This being said, for banking, insurance and other corporate sites, it's better to display the complete real name since it's based on legal documentation and information is protected, so nobody else by the user will see it