Yes, you can access files via a webpage/browser, there are also RESTful APIs for accessing data.
And yes, you can also sync p2p without data being in the cloud. For now, however, Live Mesh doesn't have a way in the UX to specify that a file shouldn't be copied to the cloud. So in the current release, all files will be copied to the cloud until you run
out of quota. At that point, files will sync p2p but not through the cloud.