Call me forever a skeptic, Mike, but I'm not buying this as the solution. You may have narrowed your search, but I don't think what you found is truly the cause for such flickering.
I don't see how you could have been getting different voltage readings from different breakers. They are all connected in parallel at the bus bar, so aside from a minuscule difference between the upper and lower ends of the panel, this couldn't have been possible. All odd-numbered poles would measure identically, as would the even-numbered ones. OK, one exception might be a bad breaker with burnt contacts inside under load.
A fault in a doorbell transformer winding wouldn't have caused flickering lights either. These things draw less wattage than a light bulb, so even a bad one couldn't cause that. If anything, a fault would cause the primary windings to blow open and just stop working.
A ground and neutral touching in a single box can't do this either. If anything, the circuit feeding this box had/has an open neutral between it and the panel. In such an instance, the nicked neutral touching the box could have caused the ground wire to become a current-carrying conductor. With poor connections to the box as you stated, that could have been the cause for the flickering. Still, you'd have seen evidence of heating at the point where the neutral was touching the box.
Unless most of the building's lighting is on this one circuit, only the most extreme circumstances would have caused flickering on other circuits. Personally, I think that the lighting circuit's neutral wire has a break, likely at where it connects to the neutral bar in the panel. People often over-tighten these screws, causing them to literally cut the wire or crimp it down to the equivalent of a much smaller gauge. It appears and even feels intact even when it isn't.
Let's not rule out a "double" trouble. There may be a loose neutral wire elsewhere, such as in a receptacle or (gasp) a buried splice. The neutral/ground cross that you found may have been carrying part of the the load while the faulty device is working just fine. Power is coming in but not going back out. I know you said that all of the wiring devices have been checked, but there could always be that one that was missed.
I hate to sound like a turd in the punch bowl, but I suspect that the problem is on the way to being fixed yet not completely. You are definitely on the right track and a faulty neutral is often the first place to look.
