Why would entropy of a system be fixed if it can exchange energy with its environment? Entropy maximization and energy minimization are equivalent statements of the same thing, as I understand it. If the internal energy is fixed, entropy is maximized because of statiatical reasons. If the entropy is fixed, and the system can exchange energy with its environment, then the system will give energy to its environment to maximize the environments entropy (and hence total entropy).
But I haven't been able to understand why we would assume here that the entropy of a system is fixed. What physical mechanism causes this?
 A: I think you are misunderstanding a few things. Both the entropy maximization principle and energy minimization principle are due to statistical reasons. Suppose we have some parameter of the system $X$ (e.g. volume, particle number, etc.) that is free to vary, and the energy is fixed at $U_0$. The system will evolve to a $X_0$ such that the entropy is maximized at the value $S_0$.
Now the energy minimization principle refers to this same point $X_0, S_0, U_0$. If we change $X$ away from $X_0$ while keeping $U_0$ fixed, $S$ will decrease. So if we want to bump $S$ back up to $S_0$ we need to add energy. So any point $X$ which has the same entropy $S_0$ must have at least as much energy as $U_0$. This is all the energy minimization principle is.
This is just an alternate way of characterizing the same point $X_0, S_0, U_0$. Again, if we move away from it while keeping $S_0$ the same, we need to add energy. It is a choice we make, not some physical mechanism.
A: The energy maximisation or equivalently entropy maximisation is a property of an equilibrium state. So in the case of when there is energy exchange with the surrounding, clearly the system isn’t in equilibrium. Once all the exchange is complete, the energy/entropy is maximised. 
