Your girl looks a lot like our Mouse bud.
Different coat, but the same dark snout.
As to cats, one of our cats seems determined to get himself killed to be honest. He's a completely a-hole! In fact, my sister (who we re-homed him from) admitted recently that that is why she gave him to us! He's a bully with other animals. They try to be nice to him, he tries to wallop them with his claws from out of reach.... When they were tiny pups, he used to walk into the living room when they had been fed, and try to intimidate them into walking away from their food so he could steal it.... Our other cat however they get on fine with, they sniff each other's behinds in greeting, then just get on with their own business... They don't get left with the cats unsupervised though, we have an extra tall dog gate on the living room door for when we go out, and one of the kids sits in there with them.
We seem to have been lucky though to be honest, apart from an abiding hatred of foxes (yeah, we have a LOT of foxes where we lived.... Walkies off the lead are not possible here, not least as we also have youths in the area who are downright evil and quite criminally inclined...), they're not very Patterdale in their behaviour. Tough as anything, and play with each other as rough as you can imagine, but I've had Cavalier pups naughtier and more destructive... A breed famous for been docile and biddable...
Breeder reckons her dogs are the same though, and past litters have grown up the same way. We met mum and dad a few times, and their mum was very very calm. Dad just wanted fussies. lol
But yeah, originally bred to hunt and kill foxes on sheep farms near the village of Patterdale. I doubt very much if they're pure though, as if the story is true, they would all have to come from a very small number of dogs from one farmer, who passed away, his son then getting the dogs and breeding from them.... Which would explain why some Patterdales look nothing like ours. I am sure you've seen the ones that look a heck of a lot like they have the face of a Staffie? Once had to argue with somebody, as they insisted ours couldn't be Patterdales, as Patterdales looks more like Staffies....