What's the purpose?
The original intent was to make it easier for women to clean the perineal area without contacting a surface, and reduces genital contact with a surface.
How do I know this? My wife was on a safety committee in a past job where there was a workplace injury claim related to this topic. In our case, there was an automated toilet cover dispenser, which made the round seat code compliant.
Yes. But that standard is being enforced by laws (and perhaps other legal mechanisms) in many places in the US. See https://en.wikipedia.org/wiki/Uniform_Plumbing_Code
Rest of the world? What about the places with squat toilets? Those are common in many places in the "rest of the world", and they don't even have seats!
Here in Singapore we have both styles of toilets.