As I read your post, I guess I see three different things you're talking about. 1. His "my way or the highway" approach to things. 2. His lack of compassion for animals. 3. You're not a quitter.
1. I've only been married 6 years, so believe me, I am not expert. I do remember the first year of our marriage (we had a honeymoon baby, so lots of adjustment). I wouldn't say my husband had that attitude, exactly, but I did have to talk to him about the fact that he was really treating me like a housekeeper and a nanny. He would come home from work and want to eat and then work in the garage. On the weekends (every weekend) he wanted to go to his parents house so he could work on things there. They have 4 acres, and most of his projects were there. Part of the problem was that I didn't want to cause trouble, but I really had to talk to him about what it was that he expected a marriage to be and why he thought his behavior was acceptable. Little by little, we worked out ways to work together, compromise, etc. The goal, afterall, is to find ways for both of us to be happy. You need to talk to him about this one.
2. I was really surprise to see the number of people who thought this was a deal breaker. I don't me to sound insensitive, but I'm just not an animal lover. I did marry a man with 3 cats, and 2 of them have pass away. I didn't personally feel any loss, but I did feel compassion for what my husband was feeling. I don't think it was wrong that he didn't personally grieve the loss of your pets, but I do think it was wrong if he didn't at least show you some compassion during your grieving process.
3. I have no idea whether your marriage is a good one or not. That would be insanely presumptuous. But I really don't believe the term "quitter" applies here at all. You're looking at that as a bad word, as if a "quitter" was someone who was lazy, didn't care, didn't feel like working to make something work. Sometimes the smartest thing a person can do is look at a situation and say, "Who benefits from this situation remaining the same?" Only you can decide whether staying in this marriage is really in your best interests and in the best interests of your children. You owe it to yourself, your husband and your children to take some time to really reflect on your life. Give yourself some time to really imagine both scenarios. If you allow yourself to really think and reflect and feel, you will know what the right decision is.