As with alot of other dog breeds you will find that some people have breed Danes to be pets and other lines have been breed to maintain their instincts out of tradition. I imagine pet lines are more prevalent. Youtube it and you will see it's not uncommon
People actively use them for wild hog hunting. They have been used to hunt hogs for something like 400 years.
People 100% use them around here bur I know nothing of trials here as I have little interest and not looked into it.
Is your pyrenees mixed? A Properly bred pyrenees with good temperament typically has no interest in things like chickens, they have a very low prey drive compared to most dogs.
I properly socialized and bred lgd is generally fine left outside from an early age. Both of mine where born in...