Axonn, Shadow Stealer and Hydraxon were members of the Hand of Artakha. Because the Hand of Artakha was disbanded before the Order of Mata Nui was formed, and the Order of Mata Nui woke up the Toa Mata, which was before Mata Nui was launched, this also means they were made by the Great Beings prior to Mata Nui's launch, unlike the vast majority of species, who were made by Mata Nui.
Now my question is, are they members of species, or are they unique individuals?
The Great Beings never seemed very focused on populating the universe with anything but Matoran, which, beside the Zyglak and Kestora, are to my knowledge the only sapient species made by Great Beings and not Mata Nui. Unless their species would serve a critical role that is, which brings up the question of what that role would be. I mean, even the Makuta were made by Mata Nui.
As for why they would have been made as unique individuals...
"Over 100,001 years ago, Shadow Stealer was a member of the Hand of Artakha, the predecessor of the Order of Mata Nui. He was considered a hero before the first Toa were created, defending the Matoran building the Great Spirit Robot from various menaces to its construction.[citation needed: BEU?]"
I would figure they'd be something like test concepts for the role of the Toa as protectors, and possibly Makuta as Rahi creators, since the HoA made the Energy Hounds.
Edit: Partial answer.
"2. Hydraxon is a member of a species, not a unique being, correct?"
Perhaps all three would be considered the same species? Or why would the Great Beings make an entire species of each of them?