You ask some good questions. As for the dog, I would agree with the others and say no. I don't believe that your average pet experiences count as animal experience for the purpose of the application.
However, I am going to count my sheep experience on mine- because livestock management is different. If I had one pet sheep, no, I wouldn't. But I have thousands and thousands of hours in breeding, lambing (!), shearing, hoof care, feeding, condition scoring, vaccinating, deworming, castrating, lancing abscesses, bandaging wounds, blah blah blah. It could probably go on forever.
As for the 4-H, I 'd count it if it applies. Hey, many people's veterinary experience goes back to when they were 13. Just because you were a kid then doesn't mean that it doesn't count. What did you do for 4-H? Raise and show a steer or hog? Have a tiny breeding flock? Participate in the Little International? It's animal experience and it's difference from basic animal ownership. I was/am extremely involved in 4-H and it's just as legitimate as anything else you could put. In most areas, you can be in 4-H until you are 19.
You could put it in community involvement if you wanted too as well. It just depends on how you view things. Some people say to put any time you touched an animal on your application- that is to me a bit overkill. Can you/should you put your animal science labs down as animal experience? I don't see why not.