Yes, it's safe to use the pan again, however, it may not have non stick qualities anymore. I've gotten my favorite pan too hot and eggs now stick. The smoke was probably from the too hot oil.
When deciding whether or not to take a pet to the vet I go through my mind asking what can the vet do. It sounds like she's OK. Of course she'll walk sort of funny because that fall caused her muscles to be sore. I wouldn't take her to the vet unless she has other symptoms, such as vomiting, Cats are quite flexible and are rarely injured in a fall.
I like huevos ranchero. Lay down a flour or corn tortilla. Top with black beans and two soft eggs. Smother in ranchero sauce or salsa. You can top it with sliced avocado, too.