You are right about the Bible not saying by what process God make all the animals other than they came from the earth. To me that means I don't know the process in detail. That's all. If it were important to know the exact process God would have told us. The Bible says it contains all things that pertain to life and godliness. So if it's not in the Bible it is apparently not necessary to know.
He created everything after it's kind (genus). The feline genus has experienced evolution within the feline genus. Some species in the feline genus have disappeared, some new ones have appeared on the scene. So, yes, cats of today are different than the original cat God created, but they are still all cats of the genus feline.
According to the Bible, an oak seed may over time produce different species of oak trees, but it never made a pine tree. A pine tree, in all it's variety (species) comes only from a pine tree seed.
For the record, there is a lot more detail on how God formed and created the animals. Please notice that I just used two different words, formed and created. They are different words that mean different things. He formed their bodies from the earth because all the things necessary to form a body were already created when God created the earth itself. Our body came from dust and that's where it will go when it's all over. Before the animals, God created just the earth which had all the material necessary to form a body. No need to recreate carbon, etc. It had already been created when He created the earth, so He just needed to form it from the dirt.
If you look closely you will see that when it says God created the animals He did so by breathing the breath of life into them (i.e. soul life), which didn't exist prior to that time. He had to create life because that didn't exist when He created the earth so He had to create something that didn't exist before.
Bottom line, God formed the body from the ground and He created the life of that body when He breathed the life into the body. People are a whole other story. God originally created man with spirit. Man's body was formed from the earth (dust), He made man a living creature when He breathed into him the same life He had created for the animals (no need to reinvent the wheel), and He created man in His own image, which the Bible says is spirit. So animals are body and soul, while man is body, soul, and spirit.
It takes a very, very careful reading of the first two chapter of Genesis and a few other verses, but that is all in there. Few take the time to see the detail that's really there. I think it is because everybody already "knows" what the Bible says without actually studying it in depth.
I don't mean to confuse you with all of this. It's like trying to teach someone calculus that is not sure about trigonometry. I can give you more on the subject if you are interested.