What sounds easy in theory often turns out to rather difficult to do in practice. As an example, this definition...
For the felines for example, they would be placental carnivore mammals that have retractable claws.
...is broken right off the bat. Your definition of "felines" excludes cheetahs but includes sea otters. Keep trying ;)
Very thin ice. How do you establish whether they're related or not? What does "related" mean in the context of special creation?
morphologically similiar, who share a set of physical characteristics, which are not posessed in total by any other group.
That's a highly subjective definition; traits that are crucial to one taxonomist may be totally discounted by another. Biblical taxonomy lumps bats in with the birds, presumably because wings are a more important trait than hair or lactation. The term "pachyderm" used to comprise such disparate animals as elephants, rhinoceri, hippopotami and tapirs.
Hmmm? What's a genus? What would a group of genera be?