Question: I was trying to transcribe my cat's noises into IPA. For example, her wanting to be petted is like [njæː] or [njaː]. But purring doesn't seem to fit any existing notation, except maybe [ʜ]. Has anyone ever mapped a cat's phonetic inventory? Their vocal tracts don't seem to be much different than ours.

Answer: 
Sure, why not? The main problem you will run into is establishing contrastiveness. I strongly recommend reading ch. 4 of the Handbook of the IPA. You can hear two different cats purring at the [wiki page](https://en.wikipedia.org/wiki/Purr), so the question is whether those are two different purr phonemes (technically known as "purreme") or one with a big variation in realization? You could transcribe it as [ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆ʔ̆ɨ̥̃̆], though you might have to explain that the breve implies a duration of about 5 msc. Apart from the inconvenience of writing all this junk, there is a deeper scientific question of whether it's wrong to imply that a purr is really made up of lots of tiny segments. So this is where, in dealing with language, somebody would propose a new symbol for this newly discovered sound. You could then propose [ﮛ] to represent this sound, but I doubt it would get approved.

With real language, the IPA symbol [a] represents a wide range of physical sounds, not a precise single physical sound (that doesn't even mean anything). At a certain point, an apparent "a" is far enough back that it sounds like the range of sounds represented as [ɑ], and then you can argue that writing "a" is an error and instead "ɑ" should be used. But that's only possible because there are languages that contrast [a] and [ɑ], so we know that [a] and [ɑ] are different things. The [a] of Arabic and the [a] of English (not US English) are different, but that doesn't motivate inventing new vowel letters: because, IPA is not a system for reducing arbitrary acoustic waveforms to combinations of letters.
The hiss sounds like an [x].

The best purring sound I can make is the Spanish trill [r] with my mouth almost closed, or modulating my mouth so that it fills with air, it leaks out, repeat.