Would atmosphere floating inside an airship be counted towards the mass of the airship?
Would atmosphere floating inside an airship be counted towards the mass of the airship?

Yes, but not for buoyancy.

Does this mean that below the hydrogen cells sits an empty volume of air equal to roughly 58% of the volume of the cells?

Yes, exactly.
When calculating the lift produced by the interior of the airship, the gas with lower density compared to outside air (hydrogen) produces lift while the internal air of equal density to outside air has no influence. It simply cancels out. While hydrogen displaces air of higher density, creating lift in the process, this internal air displaces outer air of equal density. So for lift that internal air can be disregarded.
However, when maneuvering this air does count. It increases the inertia of the ship, both when changing direction and when accelerating or decelerating. To this air the airship designers even counted the outside air which is influenced by the motion of the ship. An airship in motion continuously displaces air at its bow and lets it flow together again at the stern. The air is thus given an impulse which - as the product of mass and speed - can be described as an additional mass of the airship at this speed. In the case of an elongated body, this additional mass in the direction of travel is small in relation to the mass of the displaced air, but is of the same order of magnitude in case of lateral or vertical movements. In case of rotational movements, the inertial forces and moments of the gases within the airship envelope are added. To make it easier to calculate the forces on the airship hull, the concept of virtual mass was introduced. This virtual mass depends on temperature, altitude and speed. Although this technique allows a simple mathematical description of the effect near an operating point, it can only be generally applied with additional effort. If you are interested in the actual values: NACA-Report 184 contains tables and correction factors for the calculation of virtual mass.

I ask because water in a ship's hull makes it heavy and sink.

Because this water would displace air which is lighter than water and makes the ship float. This comparison only works when air would be replacing hydrogen in the gas bags. See the empty volume in the airship as akin to an external tank which is added to a ship and submerged. Its weight would be carried by the water. The air-filled volume in an airship is needed to have space for the gas bags to expand into without disturbing the aerodynamic shape of the hull.

Why wouldn't air in an airship make it heavy and fall to earth? Or would it?

Because this air displaces air of equal density. It cancels out in the buoyancy calculation.