all the animals to the right of the salamander would have the common characteristic of... claws or nails…

Brief Explanations:

Looking at the cladogram, the animals to the right of the salamander all share the characteristic of having lungs. Salamanders have lungs, and all the subsequent animals (lizard, pigeon, mouse, chimpanzee) also have lungs as a common - trait among them. Claws or nails are not present in all of them (e.g., pigeon doesn't have typical claws or nails like a lizard), jaws are present in animals even to the left of the salamander, and fur is only present in mammals (mouse and chimpanzee, not in lizard and pigeon).

Answer:

Lungs