Since the inner workings of the ear are made of bones, does that mean that his inner ear is also coated in Adamantium?
Wolverines Adamantium skeleton is an offesnive/defensive feature that makes his attacks stronger and protects his external frame making him stronger.
Wolverine has been shown to have had Adamantium bonded to him by at least two different methods - plating bolted directly to the bone or direct injection into the bone marrow using Adamantium Beta.
We can rule out plating of the ear bones as they are tiny and plating offers no real benefit. Plus, due to their tiny size they would heal quickly.
Bone Marrow injection
We can probably rule this out scientifically. In some Marvel comics, Wolverine has been shown to have had the Adamantium injected directly into his bone marrow. But, by age two the ear bones lose their marrow cavities and are completely ossified. Since Wolverines powers only kicked in during his fathers death well after infancy, and the adamantium injections happened decades later we can rule out his ear bones being adamantium bonded.
Initially, the malleus and incus form as a single structure, and it is only later that they separate to form two separate bones. Ossification continues through the entire fetal period, and the newly formed bones also have a transient bone marrow cavity. The marrow cavity is still present at birth, in both the malleus and the incus, and with continued ossification is lost during the first two years after birth. Postnatally, first the malleus and then the incus lose their marrow spaces. UNSW Embryology Article