I know that English can be frustrating, and I don't really want to fall into an argument, here, but I am afraid you are wrong. I realize me saying that is also frustrating. Sorry.
I don't speak Estonian, and your English is great. My feedback is meant to be constructive if you want it.
Pet my dog whom only bites sometimes.
We ask these questions:
What is the verb? The verb is "to pet."
Who is "doing" the verb? That is the subject. Here, the subject is implied; the subject is you, or the reader or listener. Like, "Hey, you pet my dog right now!" It's a command, so we omit the "you."
To whom or what is the verb being "done?" That is the object. Here, the petting is being done to the dog. The dog is the object.
If I want to say something about the dog, I use whom, or him, or any other word appropriate for referencing the object of a sentence.
Pet my dog whom some guy bit yesterday.
This is correct, but not for the reason I think you thought. Even though you brought up the crazy guy, the dog is still what the verb applies to; the dog is still the object. Implied "you" is still the subject. "Whom" is the correct choice because you are referring to the dog, not the crazy guy.
The phrase, "whom some guy bit yesterday," is actually a prepositional phrase in the sentence. It acts as an adjective decorating the dog. Because the dog is the object (being pet), "whom" is the right choice.
Consider, "My friend who some guy bit yesterday walks my dog." Here, "who some guy bit yesterday," is also a prepositional phrase. It acts as an adjective in this case, decorating my friend. Since "to walk" is the verb, my friend (doing the walking) is the subject and my dog (being walked) is the object, "who" is the right choice to reference my friend (the subject) in the prepositional phrase.
As an aside, you may like the sentence more when punctuated, "My friend, who some guy bit yesterday, walks my dog." That punctuation is not technically required, though.
From the same site as earlier offered, chosen only because of search engine rank: