Yes, all of these questions are fine. We can ask "what kind" or "what kinds" in all of these circumstances. The person answering the question will not limit themselves to a single answer if you ask "what kind". Your third question is grammatically correct, but uses unlikely wording. It would normally, but not always, be stated as "What kinds of animals are you familiar with?" or "What kinds of animals do you know about?". Again, the way you stated it is fine and you would get appropriate answers, but, we tend to not think of ourselves as "knowing" a lot of animals, rather we "know about" or "are familiar with" them. Knowing implies a personal relationship which is hard to have with many animals.

kinds are used as a plural noun in formal or written English