> The author mentions a seller of kittens, all of which have had their tails bobbed to prevent then from becoming goblins.
Bob-tailed cats are really common in Japan, and the type has been known from antiquity; literally from before there was writing in Japan. The stereotypical alley cat is usually depicted as this type. Long tailed cats were sometimes considered to be dangerous spirits in disguise (猫又 nekomata) or capable of developing into those spirits over time.
The bobbing of the tails might have been a reference to a mythological origin story (like “how the leopard got its spots”) where an emperor decreed that — following a huge fire spread by a panicked cat with its tail on fire — all cats’ tails should be cut off to prevent it from happening again.
> A character in the story says all women are shape-shifters and can become foxes, and vice-versa (I have heard this elsewhere).
Kitsune (狐), or fox-spirit. Probably one of the most commonly-known yôkai (妖怪; monster, spirit, apparition, ghost) in traditional Japanese mythology. They are supposed to be able to disguise themselves as women and/or possess people. Any woman encountered after dark was a possible kitsune in disguise.
There’s at least one story that closely parallels an Irish tale about selkies, in which looking too closely at his wife’s past destroys the marriage.
Shapeshifting, particularly into human women form, is a really, really common power for yôkai in Japanese mythology. Cats are supposed to be able to shape-shift too, as are tanuki and many other animals.
> He also mentions entering a building where dishes of salt have been left on either side of the front door, but doesn’t say why.
Salt is one of the traditional protections against evil spirits in many cultures. In Japan, you use it for purification and protection after returning from a funeral, to prevent spirits from following you home or entering your house. In many places, people will leave a small pile of salt in a bowl (盛り塩 mori-shio) to one side of the door (usually the right) as a symbolic protection. You see salt-scattering in sumô matches as well.
These are all traditional beliefs. In pre-War Japan, given that state-sponsored Shintô was promoted to more than the status as a set of folk-beliefs and remnant of animism that it had from Japanese prehistory, they were probably given far more credence than they have now. Japan is very secularist in outlook, despite the integration of many little rituals of religious origin into daily life. Considering that in modern American culture you’ve still got a significant number of people who sincerely believe that the repetition of a translation of a second-century litany originally written in Greek, or invoking the appellation of a street-preacher who got nailed to a post in Roman-era Jerusalem can protect them from Satanic possession, you might want to calibrate your belief-o-meter appropriately.