1. Are we measuring everything which affects the perception of hearing? No, we're not. ... Are we measuring everything (all the physical properties) of sound waves themselves? Yes, we are.
2. If you're talking about the latter question, physical properties of sound waves, then: frequency and amplitude. If it's the former, then the question is meaningless because it doesn't necessarily have anything to do with reality.
To be honest though, your question is irrelevant. The fundamental bottom line is: We can only record what we can measure! Or conversely; if we can't measure it, then we cannot record it! Obviously then, even if there were something that we cannot measure, your question is irrelevant because that "something" cannot be recorded and therefore does not exist in any audio recording you are replaying/reproducing!!
The problem which afflicts so much of the audiophile world is that they cannot (or will not) accept that there is difference between their perception of hearing and the sound waves themselves. This leads them to the fundamentally flawed belief that their perceptions MUST effectively be properties of sound waves, and anything which challenges this belief MUST be wrong. Pretty much any fact, "fundamental bottom line", science and even simple common sense will be sacrificed in order to maintain this belief. For example; digital audio isn't digital, audiophiles are not human, the proven mathematics (upon which all digital computer technology is based) is wrong and; if there is something they experience when listening to recordings which can't be measured, it MUST be because microphones, speakers and all past and present audio recording technology is capturing and reproducing something it was not designed to capture or reproduce and which science doesn't know exists and/or how to measure it! Unfortunately, pretty much all audiophile equipment is based in part or entirely upon this confusion (between sound waves and perception) and their marketing actively promotes, reinforces and exploits it.
G