Regarding the aircraft diverting from 1g while in level flight I have a theory.
I considered it a while back but not thoroughly enough and subsequently ended up with an unfavourable answer.
Basically it involves the vertical/normal accelerometer.
Most aircraft, B-52s withstanding, have their wings aligned with the longitudinal axis, so the nose and wings of most aircraft point in the same direction.
In order to generate lift the wings must have some AOA, but because they are aligned with the longitudinal axis, the aircraft's fuselage must also have an equal AOA.
The normal accelerometer is secured in the fuselage usually under the pilot. Let us assume again that an aircraft such as an F-16, F/A-18 or F-15 is in 1g level flight at 1000 feet and at a constant speed of 300kts. The pilot suddenly selects maximum afterburner.
In this situation, remembering that the aircraft even in level flight has some positive AOA e.g. 5 degrees, the normal accelerometer will sense a component of this longitudinal acceleration and read it as an uncommanded less than 1g excursion. To compensate, the FCS will command nose-up pitch until it establishes what it senses as 1g, but which in reality is a value greater than 1g.
This would explain why the phenomenon is most pronounced at lower speeds, (longitudinal acceleration is greatest) tapers off at higher speed (longitudinal acceleration is lowest) and stops when longitudinal acceleration ceases.
However this is just a theory unfortunately 
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I think the stability axis of an aircraft is its flight path/velocity vector? In order to calculate g at the velocity vector I believe the FCS uses the pitch rate gyro, as this should decouple g from the longitudinal axis. I previously disregarded the theory, because I pictured the decoupling as a physical procedure rather than a software/mathematical event, i.e. the actual vertical accelerometer being rotated.
To cure the pitch-up due to longitudinal acceleration if my theory is correct, a longitudinal accelerometer could be used. Apparently the Super Hornet has sensors with the capability of providing longitudinal acceleration, but they aren't utilised. The Super Hornet suffers from the same acceleration pitch-up phenomenon.
So, How'd ya like them apples?
Cheers
Obi