Would a pure flying wing in straight and level flight have zero wing bending stress? In studying the NASA 247 foot wingspan solar powered Helios aircraft it seemed that evenly spacing the weight would help create a lower bending stress across its span when in flight, as gravity and lift forces would cancel each other out more evenly as compared to a single fuselage.

The Helios did have spaced pods slung underneath, but none at the ends.  The aircraft, in turbulence (perhaps an updraft) that caused its wingtips to bend upwards, started a pitch up/pitch down cycle until the wing failed.  Could more even spacing of weight on the wing reduce the possibility of this happening again?
Net wing loading (weight per unit lifting area) can't be zero in level flight.  Lift must equal weight, else the aircraft will accelerate in the direction of the net force.

The loading distribution on the Helios is designed to reduce flexing moment along the span, which is an entirely different thing.  As you note, that optimization wasn't perfect, although pitch oscillation wasn't necessarily related to the wing flex that shows in the video.