Uh, yes, I was assuming an ideal gas here, but only to show that the pressure will be determined by the volume when T is held constant. This will also hold true for other substances.
The integrals of de/T and p/TdV will not always be the same when T is variable, but their sum will be.
Regarding your first question you have to distinguish between a process, which may be irreversible, and a path in the state space of equilibrium states used for calculation of the change of S.
Usually when there is entropy production in an irreversible process, the work is also not equal to -pdV. So there is no contradiction with TdS not being dQ. Think of stirring a viscous liquid which is a form of non-volume work which gives rise to entropy production.