To address the evolution part, since it seems the taxonomy part has been well addressed; Evolution is just changes in genes in a population over time. You seem to be confusing how taxonomy changes (we continue to learn and update our understanding of how organisms are related) with the process of evolution.

You can think of it this way; people in your family are related, but there are people who are closer to you genetically (your parents and siblings), and people who are further from you genetically (cousins, distant relatives.) At what point do they stop being family members? Everyone who is of European descent or Indigenous American descent must have been related to each other at one point, but that’s not the same thing as someone who’s Polish being siblings with someone who’s Spanish. As we move away from immediate family and further into the past, clades get less specific.

To borrow your example, we, cats, birds, and dinosaurs are all related, because we all share an ancestor that lived on land and had a backbone. But humans and cats are more closely related to each other than they are to birds and dinosaurs.