Yes, #4 looks much better this way but I probably would have gotten rid of the dark area altogether and crop some of the green on the right. You want more space in the direction the pet (or person for that matter) looks. And again, rule of thirds. If you don't know what that is, google it. It is a basic rule of composition that too many people here don't seem to understand.
As for the dog with mange, I like the 3rd one enough but it would be a lot better if you could see more of his body. Meaning if he was turned some more.
To address the equipment part of your original post, I would say that the equipment does not make the photo, the photographer does. Learn to use what you have with all its limitations. Better equipment makes the work easier but not better. A badly composed image is bad whether you are shooting it with a point-and-shoot or with an expensive SLR.
I do not want to discourage you because I do like some of your photos but they seem to be accidents more than anything else. Photography is not easy but it is not that hard to learn either. Digital photography makes it easier to learn as you don't have to spend either time or money making prints. But you still need to learn the ABCs.
And you will find the ABCs of composition in a design book rather than a photo book. When I went to art school I thought I was wasting my time except for my design professor's two classes. I quit school soon after taking his second class and most of the kudos I have gotten since have been about my compositions.
Another thing I seem to see a lot of on this forum is people cropping left and right. As an old school photographer, I learned to do my cropping in camera. Another way to say that is: when I shot the picture. You should not need to crop your photos after the fact.
Learn to shoot the photo you want and you will never crop again. Honestly though, with pets, it is easier said than done. Pets are harder to pose than people
"So would I have better luck shooting as am artistic pet photographer? Or should I just put it on the back burner till I get better?"
Marketing yourself as an artistic photographer allows you to get away with not doing the commonly expected thing. It does not, however, allow you to sell a sub-par product. You may be able to get away with it for a while but, eventually, it would catch up with you. When I marketed myself this way I had the huge advantage of hanging in galleries on a regular basis. That is to say that it wasn't just me saying "I am an artist!"
I am brand new to this forum so I still don't know all its possibilities but, if you can, let me know when you post new photos and I'll be happy to let you know what I think.
By the way, no need for that red-faced smiley. You are learning and we all did at some point. Most of us are not born knowing a good photo from a bad one. And I'll give you a gold star because you seem willing to learn. And that is very important.