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How is this done? On a purely theoretical basis, on observed behaviour, or a mix of the two?

For all up weight if the a/c climbs at say 1,000 feet per minute then we're probably happy, is there a stated minimum, I seem to recall someone mentioning 300 fpm for LAA permit?

For the CoG position is there a theoretical basis such as between 25% and 35% of the Mean Aerodynamic Chord (or whatever) or do the designers test at the airfield with varying loads until the test pilot says "That felt a bit squirrelly!!"?
MAUW is determined by several factors. The first is structural and then you get to performance. Stress calculations will be done first, at the design stage, and then structural testing will be carried out on a test airframe to prove it. Performance (climb, take of run, etc.) will be tested but are mostly likely to 'be what they are' rather than affect the cerified MAUW.

C of G is defined also by a couple of things. Longitudinal stability and then by the abillity to trim. It will initially be set theoretically but will then be validated/amended by flight testing.