Starting with 1.30.2 Blizzard's servers are actually hosting the games. They will apparently find a proxy server nearest to you to reduce ping. The current PTR has had reports of very high pings though so it doesn't seem to be working as intended. That, or they don't have the proxy servers set up yet.
That is exactly what they're doing. The patch isn't even live yet.
Oh, I was under the impression they merely provided a server for players to download the maps for improved download times. So they host the running game session as well? Pretty cool to know.