You evaluated the right part of the integral (from T1 to T2) and not left part... You have to evaluate left part too from Q1 to Q2. Someone else out there to comment?
First comment I accept Q shouldn't be a derivative with respect to time. It is simply the change with respect to one state and another.
However, Ec. (1) stands under equilibrium conditions. I mean in just one state. The rest of my derivation should be fine.
I think is a good question. The Fourier law you started: Q= m Cv dT, I think is missing a dot over the Q which means is the heat flux (derivate with respect to time). The heat capacity has a subindex V which means it is a process at constant volume. However, it would still be valid to say:
Q =...