The timing belt has to be removed to replace the water pump so it usual to do both at the same time, the timing belt requires removal to take the cylinder head off, but renewal of the water pump and timing belt is not a requirement of head gasket replacement unless is it is in bad condition.
A head gasket does not require a new engine, sometimes the head may need skimming on a machine to make it level before fitting a new gasket, especially if the engine has overheated.
I cannot see why the ring could stick unless they have allowed water to enter the cylinder when removing the head and not cleaned it which would rust the ring to the bore if left for any time. They seem to be laying it on with a service as well, It certainly looks suspicious.