Do flights have to actually go round the world?

Not at all, you can do pretty much anything you want, although prices can vary hugely for the more complicated routes. Some round the world tickets will insist that you actually go round the world but others don't, so it's down to your consultant to find the route that most suits you. As a general rule most RTWs stop in at least 3 continents going via Asia or Africa, South West Pacific & the Americas (without backtracking).