It sounds like you're under the impression that people are claiming to have a general formulation of thermodynamics that gives a complete and rigorous description of cosmology, generalizing all of the 19th-century laws of thermodynamics in an appropriate way. As far as I know, nobody in the field really makes that claim.
One reason I'm pretty sure we don't have anything like this is that the first law of thermodynamics is conservation of energy, which is usually stated in terms of a global quantity that stays the same. But general relativity does not have a global, scalar, conserved measure of energy that applies to cosmological spacetimes. (It does have things like local conservation of energy-momentum, and conservation of energy in asymptotically flat spacetimes, but that's not the same thing.)
The laws of thermodynamics also refer to temperature, but there is no really satisfactory relativistic definition of temperature.
There are the laws of black hole thermodynamics, but those don't really connect to the ordinary laws of thermodynamics in a comprehensive way, and the way that time-reversal asymmetry comes in seems to me to be qualitatively different from the way it works in standard thermodynamics. Basically you get time-reversal asymmetry in the second law of black hole thermodynamics simply because you define a horizon as a surface from which you can't reach future null infinity.
So when people (including me) say that the arrow of time comes from the fact that we had a low-entropy big bang -- well, I can only speak for myself, but I say that in a loose way, not believing that there is any fully systematic underlying theory.
So when people talk about entropy of the universe, what macrostates do they have in mind? [...] Also, when we reach the supposed "heat death of the universe". I do not see any reason why we would not be able to use different macroscopic variable in which the universe would not look like as in its "heat death" state.
I don't quite understand what objections you have in mind here, though. These seem like questions that are not qualitatively different from the ones we would ask about a steam engine, and the answers would be pretty much the same, wouldn't they? Maybe you could edit your answer to spell out in more detail what you have in mind. E.g., are you worried about how to handle the time variable, so that we can say that the macrostate is the state at a certain time? (If so, then I think the answer would be that you can take any Cauchy surface that you like.)