I have never heard of "the genitive case", but I think when you study some foreign languages, you learn grammar rules and terminology that apply and bridge knowledge that relates to that foreign language study. I think when it comes to inanimate objects, we would normally call it: the car tire, the table leg, the computer monitor. You are right about a person or animal changing the usage. We would say the cat's tail, the lady's face. This would be done without knowing the rules because when you are raised with English, you learn what "sounds right". However, to a foreigner, you would have to illustrate the "rule" because it wouldn't be evident to the untrained ear. Isn't it true, that you learn more about your own language when you study a foreign language.

It's a silly and obsolete rule--feel free to use the genitive for both animate and inanimate objects. For someone to say, e.g., "the tire of the car is flat" rather than "the car's tire is flat" is just ridiculous!

Actually, the form "mountain's top" is what is called a 'descriptive genitive'. It is more to describe than show possession. There are some very strict English teachers that tell you that it is wrong, but you will see this form used a lot and I don't see anything wrong with it. Another example of a descriptive genitive would be 'a day's pay' "Russia's economy" can certainly be used because I would hardly consider the whole country of Russia to be inanimate. My opinion, of course. In the case of "the table's leg", I would prefer a compound noun - "the table leg". It just sounds better,.

I will just refer to it as the possessive case. I think you are correct in the purest sense. Most people remain unaware or have forgotten.