Who manages air traffic during Red Bull air races?
This is a fairly broad question since the races move around the world the the regulations vary heavily depending on where its occurring but ill answer for the FAA understanding that most countries treat this kind of stuff similarly. The Air race its self has its own control tower of sorts that talks directly to the pilots to clear them onto and off of the course (this seems to be #2 in your question) but local ATC will be involved for departures. 

Depending on where they intend on flying they may need to get a special flight permit, these are applied for in advance with the FSDO. I dont know if this is something the teams need to do or if this is something organizers handle. If the country has such regulations a TFR may be issued, for example this one from the french race (thanks to @mins for digging it up!). This keeps traffic out of the way so the organizers don't to worry about a Cessna 150 putting through the course @90Kts trying to "watch the race".

The planes typically depart from a local field close to the course. Here in the US they need to follow the same procedures as anyone else, if its controlled that means ground and tower coms if its uncontrolled thats CTAF announcements. 

Once in the air they will head to the race course. Since these planes are effectively VFR radio com's is not mandatory outside of controlled airspace. The tower may simply transition them outside of the terminal area to the course frequency when appropriate. 

According to this walk through the aircraft has one radio for taking to "race control" who clear them through the course. This behind the scenes video will walk you through how race control works and what they do. 

No communication occurs during the race its self, you can see in this video the pilot is mostly concentrating on breathing and flying.

Once off the course they will either loiter around if they need to run again or head back to the airport, heading back is the same as any flight, fly on over and call up the tower to request landing. 

This podcast is also a pretty great overview on the race and what it takes to be a pilot.