Hm...I asked this cause I'm statting up the yaoguai, which seems to be a generic term for demons and monsters. Apparently they include:Most of them don't technically have immortality, they have unusually long lifespans compared to mortal versions of their kind, but do not live forever.
Animals who have become immortal via taoist sorcery, or being the pet of a god, or becoming a demon who sustains their immortality by eating people
Fallen celestials who are now demons
Beings who have always been demons
Objects which have somehow become ghosts.
The one link is tha they have achieved immortality somehow
Yaoguai is the exact same term as Youkai.
Here's a bit of a chinese supernatural glossary:
-Jing - Anything which has attained intelligence, when it normally isn't intelligent, but is benign.
--Almost anything can become Jing from age alone, so it happens to particularly old animals, plants and objects. It also happens to things around concentrations of power, tools used to perform magic, or animals that have been around gods are likely to become intelligent. Magic comes before intelligence, so they tend to have powers of their own.
-Guai - Anything with an unusual form. It's the broadest term and technically applicable to even benign supernaturals, though they'd be pretty offended.
--Adding Shou(beast) to it means a monster animal, generally extremely large, or with additional limbs and heads.
- Yao - Anything which has acquired malice. Animals which have tasted human flesh and blood, weapons used in great slaughter, ghosts which have a powerful grudge or envy for the living. This also gives them great power, because feeding on humans is one of the fastest way for a monster to grow powerful(though it caps lower than those who achieve power through enlightenment)
- Ling - Anything which is spiritual in form. Ghosts, usually, but there are also nature spirits and the like.
- Gui - Ghosts, usually, but it's often used for unidentified spooks.
- Xian - Divine, ascended to heaven in some manner.
- Mo - Magical, powerfully so. Almost always refers to demonic and corrupting routes of power.
- Tao - Understanding. Usually refers to power achieved through study and enlightenment. NOT exclusive with demonic sources of power.
So onto your examples
Animals who have become immortal via taoist sorceryAlmost never happens that way. The usual process is that they acquire the Jing state first, becoming intelligent through environment or age, then study sorcery. At some point they master a humanoid transformation, and can go further from there.
, or being the pet of a god,This is becoming powerful through ambient power. they absorb a bit of divine power, becoming intelligent, then generally steal some item of power from their master on their way out. They have access to high levels of magical knowledge as well, from exposure to divine secrets.
or becoming a demon who sustains their immortality by eating peopleThis is anything that claims enough lives, a sword, a mountain, a tiger, all of these can acquire the Jing state by acquiring enough corrupt power. They usually go through the fast way of acquiring power, by eating things from a higher state of being, they grow strong and evil.
Fallen celestials who are now demonsFallen celestials are not necessarily evil. They broke the divine rules for some reason, ranging from neglect of duty, crime or illicit love affairs(divine marriages are strictly regulated and most definitely not supposed to be sexing up mortals). Which leads to them being forcibly reincarnated as a mortal or animal(the really dangerous ones are sealed away instead).
As an animal or spirit(for severe crimes), they still possess a lot of their divine powers and memories, which allows it to fast track to become Jing, and most gods know enough transformation magic to turn into a human form even after their punishment. As a human(for lesser crimes), they usually don't possess much of their memories, but the powers are there, and will slowly transform the human back towards their divine state, while challenges and misfortune would plague them until they either reincarnate for another try, or they prove themselves worthy of divinity again, and resume their memories and title.
Some try to redeem their way back to heaven by heroic deeds and good living, others revel in their freedom as a mortal, while yet others harbor a grudge over their punishment and vent their fury upon mortals.
Beings who have always been demonsVERY few of these exist. Most of these are rogue divine enforcers that ran out of control or lost to their base impulses.
Objects which have somehow become ghosts.
See above regarding Jing.
Many of them seek immortality because they are aware of how short lived their base forms are, and the fear of death haunts them. Many of them get some level of immortality, making up for encroaching age with growing power, but it's not inherent to their state.