Impedance matching an amplifier near 1 dB compression point
Q1: Does the nonlinearity of the amplifier near 1 dB point actually
  mean that the impedance is changing, and hence I have to re-optimize?


Yes, the impedance changes. Impedance matching is a concept based on the fact that the system behaves linearly. That means doubling the input signal will double all internal signals and the output signal. Also any reflections will double.

Up until the 1 dB compression point the amplifier behaves in a linear (enough) way so that the impedance matching works as expected.

However at the 1 dB compression point the output signal of the amplifier is 1 dB lower than what it should have been if it was an ideal amplifier. That output signal being lower than expected is non linear behavior and that harms the impedance matching.

You could try and tune for that specific situation (the 1 dB compression point) but even if you find such a point it will mean that at lower power signals, the impedance matching will be less optimal. You'll need to compromise in that respect, good small signal matching or best you can get large signal matching or a compromise between those two.


  Q2: Why would the SWR show an increasing lower bound near the 1 dB
  point, even after trying to re-optimize the tuner?


I am unsure why this is but I think it is a consequence of the non-linear behavior I explained above.