You could use the peltier actively to keep both sides equally warm, then measure the necessary current. I think this should give you a very good linearity.
But you are measuring the voltage not the current. The current in this circuit is negligible. An yes, you can reach the phisical maximum and its different for every type of TEC.
im not sure about which fluctuations you are speaking, but yes they should fly perfectly linear, if they dont then external effects are at fault. c is a constant.
C is a constant but the space-time isn´t. Gravity for example is bending it and the light isn´t traveling in a absulute linear line through it.
If it were like you are saying, we would be also able to build lasers with a absolute perfect divergence with no spreading beams and the dots having always the same size no matter its 1 meter or 10 000 miles far away.