Na, we know it was in circulation, hence all the "failed incarnates" (note that they weren't 'false', just 'failed')...
I concur with Lugar2. Azura wasn't holding on to Nerevar's soul, merely keeping the right information in circulation in the Mundus to prod the vessel carrying his soul in the right direction. To pull off prophecy of that magnitude, you need the right soul, in the right place, at the right time, with the right information. Azura worked on the information part of it. I suspect Uriel Septim had something to do with the place and time part.
One difference we do know of is that the souls of humans, elves and beastfolk can't be soultrapped (except with Black Soul Gems, as in Oblivion). This is actually a major plot point in the Oblivion Mages' Guild quest, so I don't think it can be dismissed as just a game mechanic. I'm not sure whether this implies any other differences between human and animal souls, but I don't recall seeing animal ghosts in any of the games, for example.
Arkay's protection of humanoid souls, and the efforts of necromancers to bypass that protection, was also a major plot point in Redguard, where a sload necromancer erected a "soul net" making it so the only place Arkay could do his work was at one of his temples. Anyone who died outside of it was caught in the net and sold on the soul market.
The absence of animal ghosts could be for a number of reasons. First off, while I do take the things that occur in the games at face value (not the specific game mechanics that have no real-world application, but the basic events themselves), the range of possible phenomena far outstrips that which occurs in games or is described in the lore (the sum of human knowledge is finite). Secondly, it could be that the only thing that holds souls back from continuing on is a sense of purpose no animal is capable of achieving.
What about the book http://www.imperial-library.info/obbooks/souls_black_white.shtml?
Does this rule out the idea that plants have souls? Or maybe they are, um, even lesser than white souls?
Not necessarily. It could be that they're just so different that the very limited knowledge available about the souls of animals, lower and higher, is simply not applicable to plant souls. I suspect this question would only be addressed in a game (and surrounding lore) that finally took us to Black Marsh in a big way.