What if my tame leopard cat doesn't come with me? How to make it stand up? And how to take notes for creatures (animals and villagers)

Right-click it with a bone.

To change the name, you need a name label. Put the name tag in the anvil. It takes experience to change the name, and then you can right-click the creature. Killing a named creature will not drop the name tag. If you aim at it within 15, a name will appear.