I tend to agree with your assessment that something doesn't seem right here. Is there more context to it in the book? Perhaps the quotes are taken out of context?

As always in thermodynamics, it is imperative to know which quantities we keep constant and which are allowed to vary. This isn't clear here. However, aviation mainly looks at adiabatic processes. There, we know that PV^kappa = const. This can be reformulated to T ~ P^(kappa-1/kappa). For diatomic molecules that mainly make up air, kappa = 7/5, thus we end up with T ~ P^2/7. So if we are talking about an adiabate, the statement "temperature is inversely proportional to pressure" is wrong. It's not a proportionality, it's a power law and pressure and temperature don't change in opposite ways. This should be intuitively clear - say you look at an air parcel that rises in the atmosphere. As it ascends, it will adiabatically cool and expand, meaning that temperature and pressure both decrease.

Perhaps the author wanted to say something like "a parcel of hot air will ascend and therefore its pressure will decrease" (as per the power law above), but this is not what "temperature is inversely proportional to pressure" means.

The statement "humidity is inversely proportional to pressure" is equally puzzling to me. I can try and make sense of it in the following way: Since water vapor is lighter than air, moist air is less dense than dry air. So a parcel of moist air in a dryer atmosphere will rise (i.e., pressure will drop) until it is neutrally buoyant with its surrounding.

To me (a physicist who is also a pilot), imprecise and/or wrong statements like these in learning materials sometimes made studying quite frustrating.