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.