Getting a clearance
If operating at a controlled aerodrome, a rescue helicopter needs a clearance just like any other aircraft. Given the quick response required, the crew will not be able to file a flightplan before the flight, so ATC does not know when a rescue flight is coming, or where it is going.
Where I work, the helicopter base has a panel with 4 buttons at the entrance to the hangar: North, East, South and West. When the helicopter is scrambled and the crew members are running to the hangar, they will press the button corresponding to the departure direction. This gives us a warning in the tower. Typically, from the time we get the warning and until they are ready for liftoff is somewhere between 2 and 5 minutes, and since we (roughly) know they departure direction, we can start planning other traffic around it (i.e. if they are departing to the north, they need to cross the runway - to the east or west they will fly through the approach path (depending on runway in use) - to the south, normally not an issue). I imagine other airport have similar systems in place, but even if we just have to wait for the pilot to call on the radio (sometimes they forget to press the button), that still gives us 30-60 seconds of notice before they are ready to go. ATC is used to dealing with rapidly changing and unexpected situations, and this is no different.
The rescue helicopter will almost always depart VFR, which is quite easy to handle from an ATC point of view. In case of bad weather (reduced visibility or low ceiling) we might need to provide additional separation to other traffic. Rescue helicopters have priority over almost all other air traffic (except emergencies and maybe a few other obscure cases), which means that we will always clear the rescue helicopter to fly in a straight line to where they need to go - regardless of normal procedures, noise abatement and other traffic. If another aircraft is in the way of the helicopter, we move that other aircaft. In your example with a 777 on final approach, it will be told to cancel approach/go around to make way for the helicopter. I have personally done so on more than one occasion, and when you tell the jet crew the reason, they don't mind at all. It's just like making room for an ambulance on the road.
VFR vs. IFR
As mentioned above, rescue helicopters will normally fly VFR. It's not that they can't fly IFR, but the problem is that landing at the rescue site (outside of an airport) cannot be done IFR, so it doesn't really make sense to fly there IFR. In case of low ceiling (clouds) forcing the crew to go IFR, they will have a problem when coming in to land: it will be impossible for them to descend VFR below the clouds. The way they typically deal with this is to fly to a nearby airport when an instrument approach (ILS etc.), fly that approach to below the clouds and then go VFR and continue to the rescue site. This detour is obviously not desired in a situations where seconds can mean the differences between life and death.
Rescue helicopters do have lower VMC minima than other flights, allowing them to fly VFR even when other flights would not be allowed due to visibility or ceiling. The advantage of helicopters is that they can fly really slow, so low visibility is not that much of an issue.
They do sometimes pick up IFR after having picked up a patient headed for the hospital, since some hospitals do have IFR approach procedures, or when returning to base after a mission. In my experience though, they will almost always fly VFR and if the weather is too poor to allow VFR, they will normally not be scrambled at all (since they would not be able to land at the rescue site). The exception is terrain following radar equipped radar helicopters that also have collision avoidance.
