Kanaya Tenmangu shrine is located just outside what used to be the gate into Hagi castle and was where the the Daimyo and official travellers would stop and pray for a safe journey, consequently it received many paintings and such as offerings.
The paintings of horses may well be an earlier version of "ema", votive plaques which were paintings of horses as a substitute for giving a real horse. The coiled snake painting is probably connected to Benzaiten.
One of the things I look for when visiting small, local shrines is the artwork.
The final photo is one of the Zuijin at the shrine. A signboard showed picture of the Buddhist Nio guardians that guarded the shrine until the Meiji Period.