"The pilot must know the center of gravity and gross weight to fly" -- True or false? I've been trained to know the calculated gross weight and center of gravity for takeoff and landing for every flight for which I have acted as pilot in command.

I've had other instructors dispute that this is not a requirement and simply being familiar enough with the aircraft to reasonably be able to estimate that you're within C.G. tolerances considering the fuel and passengers is satisfactory.

Are there any regulations that explicitly define the requirements of pre-flight weight and balance familiarization for civil airplanes requiring an AFM containing takeoff and landing distance data?

If not, surely there is some catch-all so lawyers can sue us easier; what would that most likely be based on case law? 14 CFR 91.13, the "all available information concerning that flight" clause from 14 CFR 91.103, or something else?

Note that 14 CFR 91.103(b) requires familiarization with runway lengths of intended use and takeoff and landing distance data for any flight; myself and others would argue that being able to meet that condition would require, at a minimum, familiarization with certain weather metrics and knowing the airplane's loaded gross weight.

However, I am still not seeing an explicit requirement to know a specific C.G. figure.
True. It is the pilot-in-command's responsibility to determine if the aircraft is safe for flight. 14 CFR § 91.3(a) makes the PIC directly responsible for that. 

The 2016 FAA Weight and Balance Handbook also states:

4. The pilot in command (PIC) has the responsibility
prior to every flight to know the maximum allowable
weight of the aircraft and its CG limits. This allows
the pilot to determine during the preflight inspection
that the aircraft is loaded so that the CG is within the
allowable limits