Is the Airbus inherently stable i.e. if power and trim are set and controls are neutral, can the Airbus eventually recover level flight without input, even if it stalls completely?
In Normal Law, an airbus will not stall. Close the throttles and the airbus will go nose up and SIT there, just below stall speed. It'll spend all day like that if no other inputs are made.
In ANYTHING other than normal law, an airbus can stall, will stall if it's mishandled and it'll fall out of the sky.
So, Yes, the Airbus is inherently stable. As long as the aircraft is in normal law, quite often the correct response to any issues with the flight controls is to do nothing but monitor. The plane will fly itself.
If it's in alternate law then most of the protections are gone. At this point the PF earns his or her money. There are procedures for handling most upsets, for example, if the plane loses all speed reference data so there is no way to tell how fast or slow it's going then it'll go to alternate law because the autopilot doesn't have enough data to fly the plane.
At this point, the normal response is to leave the throttles where they are and watch the artificial horizon until the speed data comes back.
Another option is to pitch up the nose to 2.5 degress and set the N1 to 85% - both will keep the plane in the air until the speed sensors recover.
which under normal conditions would have caused the computer to advance the throttles
This is one problem with the airbus, if you close the throttles the autopilot will correct the speed but it does not move the throttles, therefore, in some situations, the position of the throttle levers does not correspond with engine power.