First let me just note how utterly ridiculous it is that pilots in 2008 are learning quick and dirty 1930's engineering to better complete safety critical tasks. There are few things in the world more silly than pilots flying new airplanes into the sides of mountains because they wasted brain cells and time doing what degreed engineers and microcontrollers should be doing far better. That said:
A normally aspirated piston engine with no altitude compensator for the mixture will, at altitude, have a richer mixture. If climbing at full throttle manifold pressure is decreasing. Likewise, reducing exhaust back pressure is increasing the volumetric efficiency. Which effects are dominant? Does the fuel flow go up or down
The exact behavior of fuel flow depends on the engine controls and the references thereof. An engine with no barometric compensation may or may not have fuel flow changes with altitude, but it will certainly have performance and specific fuel consumption changes. The question can't be answered sans handwaving without a knowledge of the controls on a particular engine.
You mention backpressure which is a measurement of limited inferential use. A drop in measured backpressure usually but not always improves net power...cylinder scavenging and the design of the exhaust manifold to achieve same is the key, and these things are weakly linked to static pressure at the exhaust outlet. Backpressure as a measurement is most useful for the mechanic/engineer trying to determine if an exhaust is blocked.
Also, keep in mind that while mass flow between intake/exhaust is similar (changing 1 part in ~12 with the addition of fuel) there is a very large difference in energy between exhaust and intake port flow. Changes in atmospheric pressure at the exhaust outlet are much less dominant than changes in pressure at the intake. Designs tend to settle on greater intake area as a result.
Surely (don't call me Shirley) the volume of gas pasing through a piston engine is constant at constant RPM.
Not even close. Load has a better correlation with mass flow than engine speed. Example at the limit: An engine with a closed throttle pumps a trivial quantity of air, but a propeller in a dive can certainly spin it quite quickly. In this case the engine produces negative net power. The key here is that load depends on factors external to the engine, such as how fast you are going and what pitch your prop is at.
Even in your original example of a climbing aircraft with wide open throttle and constant RPM, the volume flow will change, though much less than the mass flow. What you are have proposed is constant volumetric efficiency, the ratio of the charge actually inducted to the ideal under static conditions. VE isn't constant with changing atmospheric pressure, though I'm afraid any short explanation of why would amount to handwaving. Your concerns about valve events are headed in the correct direction, though overlap isn't at all the only factor.
Undoubtedly a higher exhaust back pressure means the exhaust stroke soaks up energy from the engine leaving it less power at the crank. But why would that change the volumetric flow rate?
As noted, the backpressure term is dubious so I hesitate to make a generalization based on its change. However, a change in exhaust efficiency can/will change mass flow because it is, through valve events, a player in VE. Again, cylinder scavenging is the key. Trying to describe this through backpressure alone is 1930 or even earlier thinking.
Modern high performance engines achieve VE's of ~120 percent. The energy needed to fill that cylinder beyond what atmosphere alone can do comes from the burning of fuel combined with careful design.