When the reaction quotient (Q) is equal to 1, this means that the activities of the products are the same as the activities of the reactants, meaning that we will not favor a reaction in either direction (forward or backward). Therefore, the temperature becomes irrelevant as the reaction is irrelevant. The reason why temperature is a factor for the reaction involves the fact that a reaction can be either exothermic (release heat) or endothermic (require heat). If one increases the temperature for an exothermic reaction this will shift the reaction backwards and vice versa for the endothermic case. We can relate this to the same Gibbs-Helmholtz equation (see below) as well in that when the pressure is equal to the standard pressure this term drops out and temperature is also no longer a factor because the partial pressure of the substance is not changed. The Gibbs free energy relation that you describe is directly related to the Nernst equation which you can see at this helpful website.

Gibbs-Helmoltz EquationGibbs-free-energy

Good question(s). From the way you asked your questions I think you have a good grasp of the math and working with equations. What may be causing the confusion is A) what does Q mean and what physical properties are explained by it, and B) the subtle use of equilibrium in these sorts of problems?

A) What is Q. Q can take several forms, but in this case it would look very much like equilibrium constant but at non-equilibrium values. The reason for this might not have been covered in your class and involves something called "activity". This is all very cool and can be derived from quantum mechanics but is overkill at this stage. What it means is that the concentration of each species will affect the total free energy of the cell. Since the concentration of all species at standard state is 1M then Q = 1 so the correction term drops out.

So, why if you have a cell with standard (1M) concentrations at 200 C is free energy of the reaction equal to the Standard (298 K 1M) free energy? No! This is a common mistake and by looking at the equation it seems to be true since we assume that having a variable T in the equation means the equation hold for all temperatures. It doesn't, if you look back on the assumptions we made when deriving the equation, working for all temperatures was not part of the deal. What the RT Ln Q term says, in physical terms, is that the ratio of concentrations of the will shift the energy in the cell, but Q only gives us the ratio not the energy. To turn that ratio number into energy we use RT, where T is the temp of the reaction. Here is the tricky bit, if the standard free energy is measured at 289K, then the correction term at 289K works well to tell us the energy. If we use the same standard free energy we can use the same equation if the reaction is at 30C or 15C since they are still close to the standard temp of 25 C. This will give an error in the answer but still a good approximation. However, if you are doing electrochemistry for high temperature batteries and you need very accurate numbers it would be a good practice to recalculate or remeasure your "standard" energy values at higher temperatures. People who work with neurons in the body have done this since the standard conditions in the body are different than on your lab work bench. So, yes, at Q =1 the potential is temperature independent for temps close to standard conditions but not for all temperatures.

B) You mentioned Le Chatelier's principle (LCP) which is tells where a reaction will come to equilibrium. You are right that at high T the equilibrium concentrations of the species in the battery would be different than at room temperature. However, if a battery is at equilibrium, by definition the cell potential is zero. So, what LCP tells us is the concentrations of the chemicals when the battery runs out of juice, since these concentrations vary with temp so will the point when battery cops out.