For the ferromagnetic Ising model case, outside of the dimensionality, it would be fair to assume that the lattice does not matter. There is no guarantee on this though. For a dramatic example, it should be fairly clear that the physical properties of an antiferromagnetic Ising model with be...
The point seems to missed here (cgk is close). The reason why people study phase transitions (here I talk of continuous transitions) is twofold. First there is universality, essentially that some of the physical properties of the transition (such as the critical exponents, but not the transition...
Your concerns are perfectly valid. The Landau approach is a guess, as really one has no idea what the coefficients are and which ones are the largest. This depends completely on the underlying physical system. A priori all we are assuming about the system are symmetries which is why we only...
There are a few things to address here. I'll try to give you some rough ideas, then point you to some more advanced resources at the end.
First, for the Landau-Ginzburg functional to represent your physical problem, it must have the same symmetries as the underlying system. This is why the...