Re: linear amplifier design
There are mathematical definitions of the term "linearity" using superposition properties.
But since you ask especially for an amplifier:
An amplifier has linear properties (within a certain range) as long as a sine wave is amplified without any distortion. That means, it appears at the output as an amplified sine wave. Of course, this is an idal case and will never happen in practice.
Nevertheless, we speak of "linear properties" as long as the distortions (measured as THD in %) remain below some specified limits.
In principle, you are right when you assume that in linear systems the gain is constant and independent on the input level - however, the problem is that we cannot use the term "gain" anymore if distortions occur. Because, in this case we would calculate an output/input ratio from to voltages which have not the same shape !