This question is one that requires a detailed answer so here goes.
First of all, Pyrs have been bred for thousands of years to do just what your girl is doing. Every trait you mentioned is Pyr.
Now that you know that, you can make a more sound decision on "the problems" you are having, which are not problems at all but your dog just being a pyr. Do you know what she is mixed with?
All Pyrs do not guard, or at least some are not as good a guardian as others. You will also see very different pyrsonalities between dogs. They are smart, smarter than you or I, they are self thinkers, and you need to listen to your dog when she (by her actions) tries to tell you something. I would say, from what you are telling me, that Mokey has very strong guarding instincts. A Pyr that was born and bred to guard will not be happy without something to guard. In some Pyrs, their two leggeds are enough but in Pyrs with strong guarding instinct, they absolutely have to have something to guard. Now that does not have to be cattle, or sheep, or goats, or horses. Pippin, Chelle's Pyr she bought from me has very strong guardian instincts. He is content with running out to bark at the fence line, then coming back in to guard his family which includes two leggeds, guinia pigs, rabbits, and kittens plus two legged children.
Pyrs bark to announce to the world they are on guard. They bark at all they can see and hear, like butterflys at 10,000 feet!
Pyrs patrol what they consider their territory for predators. That's why they are roamers. They will establish a path and patrol it and as time goes on the area they will guard gets bigger. What do you call a Pyr off leash? GONE!
Pyrs are nocturnal. That's why she does not sleep at night. When are predators most prevailant? About 3am. They sleep mostly during the day time but are even alert then.
I could go on forever but instead I have something that you need to read. Lisa, on this list has a new pup (Penny Lane). That pup came out of Joe Gentzle's lines. That pup has a very big chance of having strong livestock guarding instincts due to the line she came out of. Joe had a kennel dog he sent to live on a farm because he was not happy. Patou was 6yo and had never guarded in his life. Read this and you will understand your dog better. It's long but an excellent article and will give you the info that you ask for, as well as making your heart swell with pride that you own one of the amazing dogs.
This is Patou's Journal
http://www.pyrenean-journal.com/patous.htmJackie