ITA. Dogs can travel miles, so she may not live near your neighborhoods, but a few neighborhoods over a family could be terribly missing her and posting lost dog signs.
Definitely take her to the vet and get her scanned. If you end up not finding her owners and can't keep her, please take her to an animal rescue that does not euthanize. If you end up keeping her I am sure over time she and your cat will adjust to each other eventually, atleast to the point where they can ignore each other.
Yes we will be taking her to our vet to at least get her scanned and checked out to make sure she is okay. We do not want her to be euthanized. She derserves a good home. I live in a very rural area. But we will be calling local shelters and vets offices to see if anyone has lost her and putting signs up farther out once I get some more made this evening.