Okay I'm going to ramble slightly about my headcanons on Satori's pets now. But to start it off, I first have to establish my thoughts on animal/beast youkai, since it's obviously relevant.
Based on what canon lore there is, what folklore I know, and just what I feel works best, this is my thoughts on the creation of beast youkai. The vast majority of beast youkai start out as just regular animals. Any animal that can survive long enough begins to become a youkai but this happens in a series of stages. First they're a normal animal. Then their intelligence begins to grow. Eventually they reach a point of having human/youkai level intellect then their magical power also begins to grow. Eventually, some time after this, they gain the ability to take on a humanesque youkai form.
So, for example, Nazrin would have just been a normal mouse before she became a nezumi (I think Nezumi just means mouse but ... I'm using it as a youkai term because I make the rules. Kitsune just means fox after all). But yeah. The vast majority of animals don't live long enough to start becoming youkai and even among those that do many do not make it to full youkaihood.
I do think that when a beast youkai that's gained their human form has a child that that child is born within the stage that they are on. However I also generally think that youkai having children is an uncommon thing. Thus the majority of all beast youkai are incredibly long-lived animals.
Now you may recall within the lore that the reason Satori has so many pets is because animals flock to her. Specifically because her mind reading abilities allow her to communicate with them. I do not think that every animal within the world of Touhou has the intelligence to actually communicate through thought. Instead these animals that flock to Satori are those within that first stage of beast youkaihood. Where they have gained intelligence but do not have the magical ability to communicate with other youkai or humans.
Due to the fact that they have intelligence, the pets are able to free roam around the palace and surrounding area without fear of harm. Satori is fairly accepting of pets of all shapes and sizes but an unwillingness to harm one another is one of the core requirements, which she can sense with her mindreading. She'll still aid a predator unwilling to put behind its nature in finding a place it belongs but it won't be around the other pets. Maybe at the side of an animal loving oni instead.
As far as her pets go, Satori has an entire menagerie. A significant majority of the population are Kasha and Hell Crows because they are the native animal population of Old Hell. Still, the name Satori Komeiji is known across Gensokyo and other places where animals dwell, thus burgeoning beast youkai of all forms make the pilgrimage to her home. All traditional pets can be found within her palace. As can the occasional rare wild animal such as tigers, bears, boars, and elephants. The least common are actually farm animals, given most do not live remarkably long lifespans.
The majority who achieve full youkaihood do not remain at the palace of the earth spirits. Instead setting off to make their own place in the world now that they are capable of taking their full form. Satori expects this, though appreciates those few who do stick around. To support with this Satori has constructed a small hamlet not too far from Chireiden which acts as a home for many beast youkai. Some permanently settle their, others just use it as a transient space until they find a lair of their own. Some, particularly more ferocious beasts, wind up crewing with the oni of the Old Capitol.
Satori does not require her pets to work. Just like Orin and Okuu, however, many desire to do so in order to pay her back. For this reason a vast majority of the work within Chireiden is divided up between the many pets. At certain times of day it's common to see an alligator waddling along with a laundry basket or a snake handing out food. Long in the past Satori did nearly all the work in the manor but now she's mostly left to administrate.
There is a great benefit to this. The palace was made to house vengeful spirits. Something that is dangerous to both humans and youkai. The animals that work in Chireiden, however, prove to be an exception. As they are partway to becoming youkai they have natural magic defenses against spirit possession just as youkai do. Yet, as they are still long-lived animals that have not become youkai yet, they cannot have their life drained from them by a spirit the way that a youkai could. Kasha are particularly resistant, even upon becoming full youkai, for this reason they work the most directly with the spirits.
Satori knows the name of every pet but at least for the vast majority she does not name them herself. Because almost all of them are animals who lived a long life prior to becoming partial youkai, most come with their own names. Satori respects their autonomy too much to attempt to change them. Though, in regards to Hell Crows and Kasha that are born around the palace, Satori has been occasionally asked to name them. One example is Fuwa, who Satori gave her name.
As a final little detail. I think that, once they both left their isolation and began to interact with Gensokyo (maybe even some time in the past before), Satori and Kasen found common ground in which to get along. They both have pet youkai animals, which I think serves as a bonding point for them. Furthermore I write my Satori as a Taoist and Kasen is a hermit. I'm not sure that they would interact regularly but I imagine they direct pets that they believe the other could better care for to each other. Maybe have some form of playdates too.
As a second final little detail. Unless someone else would also like to use this relationship, this only applies to my Satori and the Yamame I write on my multimuse but: Yamame is an example of one of Satori's pets that moved out after achieving their youkai form.
