The entropy of sub systems is additive so the overall entropy of the universe must be greater than zero. This must be equal to T dS hence dS of the whole universe must be positive not zero?

No. But that does not mean I think entropy is increasing in the idealized models of a homogeneous, isotropic universe based on the Friedmann equations. Your references make it obvious that it is not. So, if entropy is in fact increasing in our universe, it is obviously doing so as a result of some process that is not included in those idealized models.

For an isolated system like the universe, the Clausius inequality reduces to $$\Delta S\gt\int{\frac{dq}{T}}=0$$This means that, even with no heat transfer to the system (dq=0), entropy is generated with the system itself, and thereby increases.

There are no "subsystems" in the Friedmann models, or in the thermodynamics based on them that you referenced. There is just one homogeneous, isotropic universe.

For our actual universe, yes, I agree. For the idealized universe in the Friedmann models, however, you might want to rethink this statement.

No, ##T dS## is the change in entropy as a result of some process. If the process is adiabatic, as the expansion of the homogeneous, isotropic universe is in the Friedmann models, then ##dS = 0##. Your references make that clear. What they do not discuss at all is what other processes might be taking place in the actual universe (as opposed to the idealized universe in the Friedmann models) that might increase entropy.